functions and graphs intmath

7/31/2019 Functions and Graphs Intmath

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Functions and Graphs

http://www.intmath.com/functions-and-graphs/functions-graphs-intro.php

By M Bourne

Rene Descartes

A very significant development in mathematics was the introduction of theCartesian Coordinate system (orx-y coordinate system), developed byRene Descartes (1596 - 1650). We usually draw the graph of a functionusing the Cartesian Coordinate system. This system made a lot of newmathematics possible, including calculus.

The graph of a function is really useful if we are trying to model a real-world problem. Sometimes we may not know an expression for a functionbut we do know some values (maybe from an experiment). The graph cangive us a good idea of what function may be applied to the situation to

solve the problem.

In this Chapter

Functions Overview


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1. Introduction to Functions - definition of a function, function notation andexamples

2a. Domain and Range of a Function - thex- andy-values that a functioncan take

2b. Functions from Verbal Statements - turning word problems intofunctions

Graphs of Functions

3. Rectangular Coordinates - the system we use to graph our functions

The cartesian plane

4. The Graph of a Function - examples and an application5. Graphing Using a Computer Algebra System - some thoughts on usingcomputers to graph functions

6. Graphs of Functions Defined by Tables of Data - often we don't have analgebraic expression for a function, just tables

7. Continuous and Discontinuous Functions - the difference becomes

important in later mathematics8. Split Functions - these have different expressions for different values ofthe independent variable

9. Even and Odd Functions - these are useful in more advancedmathematics
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Let's now learn about definition of a function and function notation .

1. Introduction to Functions

In everyday life, many quantities depend on one or more changingvariable quantities eg:

(a) plant growth depends on sunlight and rainfall

(b) speed determination depends on distance travelled and time taken

(c) voltage depends on current and resistance

(d) test marks depend on attitude, listening in lectures and doing tutorials(among many other variables!!)

Functions

A function is a rule that relates how one quantity depends on otherquantities. For example,

(a) V = IR where

V= voltage (V)

I= current (A)

R = resistance ()

IfIincreases, so does the voltage (assuming resistance is constant).

IfR increases, so does the voltage (assuming current is constant).

(b) where

s = speed (m / s)

d= distance (m)
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t= time taken (s)

Ifdincreases, the speed goes up (assuming time is constant).

Iftincreases, the speed goes down (assuming distance is constant).

Definition of a Function

Whenever a relationship exists between two variables (or quantities) suchthat for every value of the first, there is only one corresponding value of thesecond, then we say:

"The second variable is a function of the first variable."

The first variable is the independentvariable (usuallyx), and the second

variable is the dependentvariable (usuallyy).

The independent variable and the dependent variable are real numbers.

Example 1

We know the equation for the area of a circle from primary school:

A = r

2

This is afunction as each value of the independent variable rgives us onevalue of the dependent variableA.

General Cases

We usex (independent) andy (dependent) variables for general cases.

Example 2

In the equation

y = 3x + 1,

y is a function ofx, since for each value ofx, there is only one value ofy.

If we substitutex = 5, we gety = 16 and no other value.


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The values ofy we get depend on the values chosen forx.

Therefore,x is the independentvariable andy is the dependentvariable.

Example 3

The forceFrequired to accelerate an object of mass 5 kg by anacceleration ofa ms-2 is given by:F = 5a.

Here,Fis a function of the acceleration, a.

The dependentvariable isFand the independentvariable is a.

Function Notation

We normally write functions as:f(x) and read this as "functionfofx".

We can use other letters for functions. Common ones areg(x) and h(x). Butthere are also ones likeP(t) which could indicate powerat time t.

Example 4

We often come across functions like:y = 2x2+ 5x + 3

We can write this using function notation:

f(x) = 2x2 + 5x + 3

Function notation is all about substitution.

The value of the functionf(x) whenx = a is written asf(a).

Example 5

If we havef(x) = 4x + 10, the value off(x) forx = 3 is written:

f(3) = 4 3 + 10 = 22

Whenx = 3, the value of the functionf(x) is 22.


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Mathematical Notation

Mathematics is often confusing because of the way it is written.

We write 5(102) and it means 5 102 = 500.

But if we write a(102), this could mean

"function a of 102" (that is, the value of the function a when theindependent variable is 102) or it could mean

a 102 = 100a.

You have to be careful with this.

Also, be careful when substituting letters or expressions into functions.

See a discussion on this: Towards more meaningful math notation.

Example 6

Ifh(x) = dx3 + 5x then value ofh(x) forx = 10 is:

h(10) = d(10)3 + 5(10)

= 1000d+ 50

Example 7

If the height of an object at time tis given by

h(t) = 10t2 2t, then

a. The height at time t= 4 ish(4) = 10(4)2 2(4) = 10 16 8 = 152

b. The height at time t= b is

h(b) = 10b2 2b

c. The height at time t= 3b is
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h(3b) = 10(3b)2 2(3b) = 10 9b2 6b = 90b2 6b

d. The height at time t= b + 1 is

h(b + 1) = 10(b + 1)2 2(b + 1)

= 10 (b2 + 2b + 1) 2b 2

= 10b2 + 20b + 10 2b 2

= 10b2 + 18b + 8

Exercises:

Evaluate the following functions:(1) Givenf(x) = 3x + 20, find

a.f(-4) b.f(10)

Answer

a. f(-4) = 3(-4) + 20 = -12 + 20 = 8

b. f(10) = 3(10) + 20 = 30 + 20 = 50

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(2) Given that the height of a particular object at time tis

h(t) = 50t 4.9t2, find

a. h(2) b. h(5)

Answer

a. h(2) = 50(2) 4.9(2)2 = 100 19.6 = 80.4

b. h(5) = 50(5) 4.9(5)2 = 250 122.5 = 127.5

(3) The voltage, V, in a circuit is a function of time t, and is given by:
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The domain of the function isx 4, sincex cannot take values less than4. (Try some values in your calculator, some less than 4 and some more

than 4. The only ones that "work" and give us an answer are the onesgreater than or equal to 4).

Note:

1. The enclosed (colored-in) circle on the point (-4, 0). This indicatesthat the domain "starts" at this point.

2. Don't worry about where that graph came from - we'll learn how todraw these later, in section 4, Graph of a Function.

Range

The range of a function is the complete set of all possible resultingvalues of the dependent variable of a function, after we have substitutedthe values in the domain.

In plain English, the definition means:

The range of a function is the possibley values of a function that resultwhen we substitute all the possiblex-values into the function.

When finding the range, remember:

Substitute differentx-values into the expression fory to see what ishappening

Make sure you look forminimum and maximum values ofy Draw a sketch!
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Example 1: Let's return to the example above,y = (x + 4).

We notice that there are only positivey-values. There is no value ofx thatwe can find such that we will get a negative value ofy. We say that therange for this function isy 0. (The squiggle at the top of the arrow

indicates the range goes on forever, beyond what is shown on the graph.)

Example 2: The curve ofy = sinx shows the range to be betweeen 1 and1.

The domain of the functiony = sinx is "all values ofx", since there are norestrictions on the values forx.

More Domain and Range Examples

You can see more examples of domain and range in the section InverseTrigonometric Functions.
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Note 1: Because we are assuming that only real numbers are to be usedin the domain and range of a function, values that lead to division by zeroor to imaginary numbers are not included. The Complex Numberschapter explains more about imaginary numbers.

Note 2: Many people ask for the square root example, "What about thenegative values when we find a square root?" A square root has at mostone value, not 2. See this discussion: Square Root 16 - how manyanswers?

Also, we are talking about the domain and range offunctions, which haveat most oney-value for eachx-value.

Exercise 1

(a) Find the domain and range for the functionf(x) =x2 + 2.

Answer

The function

f(x) =x2 + 2

is defined for all real values ofx (because there are no restrictions on thevalue ofx).

Hence, the domain off(x) is

"all real values ofx".

Sincex2 is never negative,x2 + 2 is never less than 2

Hence, the range off(x) is

"all real numbersf(x) 2".

We can see thatx can take any value in the graph, but the resultingy = f(x)values are greater than or equal to 2.
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Note

1. It is important to label the axes when sketching graphs. It helps withunderstanding what the graph represents.

2. We'll learn how to sketch such graphs later, in Graph of a Function.

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(b) Find the domain and range for the function

Answer

The function

is not defined fort= -2, as this value would result in division by zero.

Hence the domain off(t) is

"all real numbers except -2"
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Also, no matter how large or small tbecomes,f(t) will never be equal tozero.

So the range off(t) is

"all real numbers except zero".

We can see in the graph that the function is not defined fort= -2 and thatthe function (they-values) takes all values except 0.

Exercise 2

Find the domain and range for the function

Answer

The function

is not defined for real numbers greater than 3, which would result inimaginary values forg(s).

Hence, the domain forg(s) is "all real numbers,s 3".

Also, by definition, 0.
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Hence, the range ofg(s) is "all real numbersg(s) 0"

We can see in the graph thats takes no values greater than 3, and that therange is greater than or equal to 0.

How to find the domain

In general, we determine the domain of each function by looking for those

values of the independent variable (usuallyx) which we are allowed touse. (Usually we have to avoid 0 on the bottom of a fraction, or negativevalues under the square root sign).

The range of each function is found through an inspection of the function.(What are the resultingy-values?)

Exercise 3

Find the domain and range for the function defined as

f(x) =x2 + 4 forx > 2

Answer

The functionf(x) has a domain of "all real numbers,x > 2" by definition.

To find the range:


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whenx = 2,f(2) = 8 whenx increases from 2,f(x) becomes larger than 8

Hence, the range is "all real numbers,f(x) > 8"

Here is the graph of the function, with an open circle at (2, 8) indicatingthat the domain does not includex = 2 and the range does not includef(2)= 8.

The function is part of a parabola. [See more on parabola.]

Exercise 4

More Domain and Range Examples

In case you missed it earlier, you can see more examples of domain andrange in the section Inverse Trigonometric Functions.

We are told that the height h, in metres, of a certain projectile as a functionof time t, in seconds, is

h = 20t 4.9t2

Find the domain and range for the function h(t).

Answer

Generally, negative values of time do not have any meaning. Also, weneed to assume that the projectile hits the ground and then stops - it doesnot go underground.
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So we need to calculate when it is going to hit the ground. This will bewhen h = 0. So we solve:

20t 4.9t2 = 0

Factoring gives:

(20 4.9t)t= 0

This is true when

t= 0 s,

or

t= 20/4.9 = 4.082 s

Hence, the domain of the function h is

"all real values oftsuch that 0 t 4.082"

We can see from the function expression that it is a parabola with itsvertex facing up. (This makes sense if you think about throwing a ball toyour friend. It goes up to a certain height and then falls back down.)

What is the maximum value ofh? We use the formula for maximum (orminimum) of a quadratic function.

The value oftthat gives the maximum is

t= -b/2a = -20/(-2 4.9) = 2.041 s

So the maximum value is

20(2.041) 4.9(2.041)2 = 20.408 m

By observing the function ofh, we see that as tincreases, h first increasesto a maximum of 20.408 m, then h decreases again to zero, as expected.

Hence, the range ofh is

"all real numbers, 0 h 20.408"
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Here is the graph of the function h:

Functions defined by coordinates

Sometimes we don't have continuous functions. How do we find thedomain and range? Let's look at an example.

Exercise 5

Find the domain and range of the function defined by the coordinates:

{(4, 1), (2, 2.5), (2, 1), (3, 2)}

Answer

Here is the graph of our discontinuous function.


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The domain is simply thex-values given:x = {4, 2, 2, 3}

The range consists of thef(x)-values given:f(x) = {1, 1, 2, 2.5}

2b. Functions from Verbal Statements

Example 1:

The fixed cost for a company to operate a certain plant is $3,000 per day.It also costs $4 for each unit produced in the plant. Express the daily costCof operating the plant as a function of the numbern of units produced.

Answer

The daily total cost Cequals the fixed cost of $3,000 plus the cost of

producing n units.

Since the cost of producing 1 unit is $4, the cost of producing n units is$4n

Thus the total cost C, where C=f(n) is

C= 3000 + 4n
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Here the domain is "all integer values, n 0" while the range is "all integervalues, C 3000"

Example 2:

An architect designs a window such that it has the shape of a rectanglewith a semicircle on top, as shown. (This kind of window is called aNorman window).

The architect wants the base of the window to be 10 cm less than theheight of the rectangular part.

Express the perimeterp of the window as a function of the radius rof thecircular part.

Answer

We label the points on the window for convenience:
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Perimeterp, wherep =f(r):

p = circumference of the semicircle + length AB + length BC + length CD

p = (2 r) + (2r+ 10) + 2r+ (2r+ 10)

p = r+ 6r+ 20

Since the radius cannot be negative, and there would be no window ifr=0, then...

The domain of the function is: all real values, 0 < rR, whereR is somemaximum possible value ofr(determined by design considerations).

Exercises

Question 1

ForF(t) = 3t- t2 fort 2, findF(2) andF(3).

Answer

F(t) = 3t- t2

F(2) = 3(2) - (2)2

= 6 - 4

= 2

F(3) is not defined since t 2.

Question 2

A rocket burns up at the rate of 2 Mg/min after falling out of orbit into theatmosphere. If the rocket weighed 5,500 Mg before re-entry, express itsweight w as a function of the time t, in minutes, of re-entry.

Answer
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Note: 1 Mg = 1 megagram = 1 million grams = 1000 kg

Answer: w = 5500 - 2t

3. Rectangular Coordinates

A good way of presenting a function is by graphicalrepresentation.

Graphs give us a visual picture of the function.

There are several different ways to graph a function, but the most commonway is to use the rectangular co-ordinate system. This consists of:

Thex-axis;

They-axis;

The origin (0,0); and

The four quadrants, normally labelled I, II, III, IV

Normally, the values of the independent variable (generally thex-values)are placed on the horizontal axis, while the values of the dependentvariable (generally they-values) are placed on the vertical axis.

Thex-value, called the abscissa, is the perpendicular distance ofPfromthey-axis.

They-value, called the ordinate, is the perpendicular distance ofPfromthex-axis.


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The values ofx andy together, written as (x,y) are called the co-ordinatesof the pointP.

Example 1

Locate the points A(2 , 1) and B(-4 , -3) on the rectangular co-ordinatesystem.

Answer

To answer this properly, we need to do the following:

1. Label the axes withx andy.2. Put a scale on the axes (the numbers) such that the points will fit on

the graph.3. Then put dots for the required points A and B.

Here is our result.

Example 2
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Three vertices of a rectangle are A(-3 , -2), B(4 , -2) and C(4,1). Where isthe fourth vertex D?

Answer

Here are the positions of points A, B and C:

Since the opposite sides of a rectangle are equal and parallel, we can seethat:

they co-ordinate of D is 1

thex co-ordinate of D is -3

Hence, the co-ordinates of D are (-3, 1).

Easy to understand math videos: MathTutorDVD.com

Example 3

Where are all points (x ,y) for whichx < 0 andy < 0?

Answer

We have:

x < 0 means thatx is negative, andy < 0 also means thaty is negative,
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So the only region where both co-ordinates for all points are negative isthe "third quadrant (III)".

The shaded area represents the region in question:

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Exercises

Q1 Where are all the points whose abscissas equal their ordinates?

Answer

"Abscissas" meansx-values, while "ordinates" meansy-values.

So the question means "where on the rectangular system do we havex =y for all points (x,y)?"

In other words, we want a line connecting points like (-3, -3) and (0, 0) and(5, 5) and (700, 700).

The line we want cuts the first and third quadrants in half at 45. We canwrite this line asy =x.


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Q2 Where are all the points (x,y) for whichx = 0 andy < 0?

Answer

They are on the negative part of they-axis.

4. The Graph of a Function

The graph of a function is the set of all points whose co-ordinates(x,y) satisfy the functiony =f(x). This means that for eachx-value there isa correspondingy-value which is obtained when we substitute into theexpression forf(x).

Since there is no limit to the possible number of points for the graph of the

function, we will follow this procedure at first:

- select a few values ofx- obtain the corresponding values of the function- plot these points by joining them with a smooth curve

However, you are encouraged to learn the general shapes of certaincommon curves (like straight line, parabola, trigonometric and exponentialcurves) - it's much easier than plotting points and more useful for later!


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Example 1

A man who is 2 m tall throws a ball straight up and its height at time t(ins)is given by h = 2 + 9t- 4.9t2 m. Graph the function.

Answer

We start at t= 0 since negative values of time have no practical meaninghere.

t 0 0.5 1 1.5 2

h 2 5.3 6.1 4.5 0.4

This shape is called a parabola and is common in applications ofmathematics.

NOTE:

(1) This graph is height against time. The ball went straight up, notforward. (Our graph may give the impression the ball moved in thex-direction as well as up, but this was not the case.

(2) We could have written the function in this example with h(t) rather thanjust h as follows: h(t) = 2 + 9t- 4.9t2. They mean the same thing.


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Example 2

The velocity (in m/s) of the ball in Example 1 at time t(ins) is given by

v = 9 - 9.8t

Draw the v-tgraph. What is the velocity when the ball hits the ground?

Answer

Since we recognise it is a straight line, we only need to plot 2 points andjoin them. But we find 3 points, just to make sure we have the correct line.

t 0 1 2

v 9 -0.8 -10.6

Our graph starts at t= 0 (since negative time values have no meaning in

this example).


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For the first 0.918 s, the ball us going up (positive velocity), but slowingdown.

Thereafter, the ball is coming down towards the ground.

The ball hits the ground at approx t= 2.04 s (we can see this fromExample 1). The velocity when the ball hits the ground from the graph we

just drew is about -11 m/s. The graph stops at this point.

Our graph assumes the ball lands in sand and doesn't bounce.

Normally, as we have done here, we take velocity in the up direction to bepositive.

Example 3

Graph the functiony = x - x2

Answer

(a) Determine the values in the table

x -2 -1 0 1 2 3

y -6 -2 0 0 -2 -6

(b) Sincey = 0 for bothx = 0 andx = 1, check what happens in between.

That is, for , we find that .


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Note the curve continues beyond what is shown in the graph.


Example 4

Graph the function

Answer

(a) Note:y is not defined forx = 0, due to division by 0

Hence,x = 0 is not in the domain

(b) Draw up a table of values:

x -4 -3 -2 -1 1 2 3 4

y 3/4 2/3 1/2 0 2 3/2 4/3 5/4

(c) Check what happens betweenx = -1 andx = 1:
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when ,y = -1

when ,y = 3

(d) Asx gets closer to 0, the points get closer to they-axis, although theydo not touch it. They-axis is called an asymptote of the curve.

There is another aymptote in this curve:y = 1. Notice the curve does notpass through this value.


Example 5

Graph the function

Answer
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(a) Note:y is not defined for values ofx less than -1

Hence,x < -1 is not in the domain

(b) determine the values in the table:

x -1 0 1 2 3 4 5

y 0 1 1.4 1.7 2 2.2 2.4

(c) The positive square root is indicated,hence, the range consists of allpositive values ofy, including 0 (ie.y 0)


Example 6

The electric power P (in watts) delivered by a battery as a function of theresistance R (in ohms) is :

Plot the power as a function of the resistance.

Answer
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(a) Negative values forR have no physical significance, hencePis notplotted for negative values ofR.


R 0 1 2 3 4 5P 0 44.4 32.0 24.5 19.8 16.5

(c) Check what happens betweenR = 0 andR = 1 ie.

whenR = 0.25,P= 44.4

R = 0.50,P= 50

R = 0.75,P= 48

Note the axes are labelled withR (resistance) andP(power).

(d) Conclusions:

(i) The maximum power of 50 W occurs when resistanceR = 0.5 W

(ii)Pdecreases asR increases beyond 0.5 W


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Exercises

Graph the given functions

Q1y = x3 x2

Answer

(a) There are no restrictions on the values thatx can take in this example.


x -1 0 1 2 3

y -2 0 0 4 18

Sincey = 0 whenx = 0 andx = 1, we examine what happens betweenthose 2x-values:

Whenx = 1/2,y = -1/8.

Here is our graph:


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Q2

Answer

We can only take the square root of a positive number sox 0. Thesquare root of a number can only be positive, soy 0.

This graph is actually one half of a parabola, with horizontal axis.


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Conical water tank

Q3. (Application) Water flows out of an inverted cone (ie the water flowsthrough the pointy end of the cone and the widest part of the cone is at thetop). The volume of the water is decreasing at a constant rate.

Draw a sketch graph of the height of the water in the cone versus the time.

Answer

We need to model the height at time tbased on what we know aboutcones. We also need to assume several things.

The volume of a cone is V= r2h/3.


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For simplicity, let r = h, then V= h3/3.

So the height of the cone is given by the cube root of3V/.

We take a cone with "easy" values, say h = r = 10. This has volume1000/3 units.

If the water drains out in 10 seconds, it means 100/3 units will drain outeach second. Thus the amount of water left aftertseconds is given by(1000 100t)/3 units.

The height at time twill be the cube root of(1000 - 100t).

That is,

The graph of our model is given below.

5. Graphs Using a Computer Algebra System


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The graph ofz= 4x3 +y3

We now have many tools which can help us to do mathematics.

All of your mathematics subjects require you to draw graphs. It is good todo lots of practise. Another way is to get some useful tools.

Once you start using a Computer Algebra System you will find that itchanges your whole view on mathematics. It takes a lot of the tedium outof algebra and frees up time to think about what you are doing.

Some of the many commercial CASs available:

Matlab - an industry leader MathCad - a sophisticated CAS Mathematica - also very sophisticated and also has a Web output Scientific Notebook (used in Interactive Mathematics. See SNB

Information)and the related product Scientific Workplace LiveMath (used in Interactive Mathematics - see LiveMath Highlights Derive (now part of Texas Instruments)

Open Source Computer Algebra Systems

For those of you on a tight budget (aren't we all?), here is a list ofopensource computer algebra systems.

Graphics Calculators

I was actually never in the position where I needed to use graphicscalculators. They are certainly better than nothing and make class sets

feasible.
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Online Graphing Calculator: Plot your own SVG Math Graphs

You can plot 2 functions, function 1 (in blue) and function 2 (in green).Edit your functions and then click the "Graph it" button below.

Function 1:Function 2:

Minx-value: Maxx-value:

Miny-value: Maxy-value:

Grid space (vertical): Grid space (horizontal):

Some graphs to try

The graphing calculator will accept any of the following functions (use thenotation shown):

Straight lines: (like 3x - 2) Polynomials: (like x^3 + 3x^2 - 5x + 2) Any of the trigonometric functions: sin(x), cos(x/2), tan(2x),

csc(3x), sec(x/4), cot(x) The inverse trigonometric functions: arcsin(x), arccos(x),

arctan(x), arccsc(x), arcsec(x), arccot(x) Exponential (e^x) and logarithm (ln(x) for natural log. For log base

10 you need to put ln(x)/ln(10). I'm using Change of Base rule.)

Absolute value: use "abs" like this: abs(x) The hyperbolic functions and their inverses: sinh(x), cosh(x),tanh(x), arcsinh(x), arccosh(x), arctanh(x)

Sign (returns 1 if the sign of the function value is positive, 1 if thesign of the function is negative. For example, try sign(sin(x)))

In fact, you can use most of the javascript math functions, including

ceiling: ceil(x) and round: round(x) random: random(x);
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square root: sqrt(x)

You can also use any combinations of the above, like "ln(abs(x))".

You can just have one graph if you want - just leave one of the "function"boxes empty.

More Information

The above illustration is an example of vector graphics - images that aremade using vectors.

The graph is not an image (that is it's not a .GIF or .JPG). It is a scalablevector graphics text file which is being rendered by your browser as apicture.

Credit to the developers ofASCIISVG, particularly Peter Jipsen, ChapmanUniversity.

See also: Math graphs on the Web without images

For more information about vectors, see:

Vector Addition (example graphs showing many examples of vector

addition) Vector Art (an important new branch of art, made using vectors)

6. Graphs of Functions Defined by Tables of Data

One important way to show the relationship between variables is by usinga table of values obtained by observation and experimentation.

Such data values would indicate whether the variables are related (ie.have a formula that links them).

Such points are normally plotted with a smooth curve.

Exception:
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When data values are taken only for certain intervals or are averaged overthe intervals, then the intervals between the points have no real meaning.

Such points are connected by straight line segments.

Example 1

The electric energy usage (in MJ) for a particular house for each month ofa certain year is given in the following table:

Month Jan Feb Mar Apr May Jun

Energy Usage 10 504 12 363 10 168 7 500 4 825 3 568

Month Jul Aug Sep Oct Nov Dec

Energy Usage 2 548 2 887 3 301 5 748 7 302 9 706

Plot these data

Answer

Since we are given the total energy usage for each month, there is nomeaning to the intervals between the months. Therefore, a histogram isused. We don't join the data points with lines.


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Example 2

Steam in a boiler was heated to 150 C. Its temperature was then recordedeach minute as follows:

Time (min) 0.0 1.0 2.0 3.0 4.0 5.0Temp(C) 150.0 142.8 138.5 135.2 132.7 130.8

Plot the graph.

Answer

Since the temperature changes in a continuous way, there is meaning to

the values in the intervals between the points. Therefore, the points arejoined by a smooth curve.

The "squiggle" on they-axis indicates the scale does not start at 0.

Estimating values

We can estimate values of one variable for given values of the other.

For instance, the temperature after 2.5 min can be estimated from thegraph as 137 C (dark red arrows below).


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Using the method of proportions:

Sox = -1.7

Therefore

142.8 - 1.7 = 141.1 C is the required temperature

Exercise

The following table gives the fractionfof the total heating load of a systemthat will be supplied by a solar collector of areaA:

A(m2) 20 30 40 50 60 70 80

f 0.22 0.30 0.37 0.44 0.50 0.56 0.61

By means of linear interpolation, forA = 36 m2, findf.

AnswerWe use one portion of the table and add our required unknown, as follows:

A(m2) 30 36 40

f 0.30 ?? 0.37

Using proprtions again, we have:


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Solving forx gives:

x = 0.042

So the required value isf= 0.37 0.042 = 0.328

7. Continuous and Discontinuous Functions

by M. Bourne

This section is related to the earlier section on Domain and Range of aFunction. There are some functions that are not defined for certain values

ofx.

Continuous Functions

Consider the graph off(x) =x3 6x2 x + 30:

We can see that there are no "gaps" in the curve. Any value ofx will giveus a corresponding value ofy. We could continue the graph in the negativeand positive directions, and we would never need to take the pencil off thepaper.

Such functions are called continuous functions.

Functions With Discontinuities

Now consider the function .
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We note that the curve is not continuous atx = 1.

We observe that a small change inxnearx = 1 gives a very largechange in the value of the function.

For a function to be continuous at a point, the function must exist at thepoint and any small change inx produces only a small change inf(x).

In simple English: The graph of a continuous function can be drawn

without lifting the pencil from the paper.

Many functions have discontinuities (i.e. places where they cannot beevaluated.)

Example

Consider the function . Factorising the denominator gives:

.

We observe that the function is not defined forx = 0 andx = 1.


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We see that small changes inx near 0 (and near 1) produce large changesin the value of the function.

We say the function is continuous for all values ofx exceptx = 0 andx =1.

Note: You will often get strange results when using Scientific Notebook (orLiveMath or any other mathematics software) if you try to graph functions

which have discontinuities. Here is the same function in thedefault graph view in Scientific Notebook:

It is showing us all the vertical values that it can (from an extremely smallnegative number to a very large positive number) - but we need to restrictthose values so we can see the true shape of the curve, like this (I have

changed the view of the vertical axis from -12 to 10):


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Continuity and Differentiation

Later you will meet the concept of differentiation. We will learn that afunction is differentiable only where it is continuous

8. Split Functions (Piecewise-defined functions)

By D Hu and M Bourne

Most functions you are familiar with are defined in the samemanner for all values ofx. However, there are some functions which aredefined differently in different domains. These are known as split

functions (or piecewise-defined functions).

Because split functions may have drastically different behaviours indifferent domains (that is, for differentx-values), it is quite common for asplit function to be non-continuous (and as we learn later, it cannot bedifferentiated).

Example 1 - Ordinary Function for Comparison

f(x) = -x2 + 4
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This function is not a split function. It is defined the same way for all valuesofx. To find the value of the function at a givenx-value, simply substituteintof(x) = -x2 + 4

Some values forf(x) = -x2 + 4 are as follows:

x -3 -2 -1 0 1 2 3

f(x) -5 0 3 4 3 0 -5

Example 1 - Split Function

In the regionx < 1, we have a straight line with slope 2 andy-intercept 3.Asx approaches 1, the value of the function approaches 5 (but does notreach it because of the "


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This function has a discontinuity atx = 1, but it is actually defined forx = 1(and has value 1).

Here's how it looks in LiveMath:

LIVEMath

Later we will learn about Differentiation. This function is differentiable forall values ofx exceptx = 1.

Example 2

Answer

In the regionx< -2, the function is defined as:

y = -2x - 8

Asx gets closer to -2 from the left side, we can see that the value of thefunction gets closer to -4.
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Now forthe regionx> -2. The function on this side is defined as

y = 3x + 2

Asx approaches -2 from the right, we see that the function value alsoapproaches -4.

The function is not defined atx = -2 so it is not continuous there. Werepresent this with an open circle on the graph.


Example 3
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This function is split into two pieces.

Fornegative values ofx, the function is identical tox (straight line).

Fornon-negative values ofx, the function is identical to 1/5 sin 5x. Again,the function is defined for all values ofx. However, in this case, thefunction is continuous (and differentiable) everywhere.

x -2 -1 0 1 2

f(x) -2 -1 0 -0.192 -0.109

Some split functions are so commonly used that they are given specialnotation.

Example 5 - Absolute Value Function

f(x) = |x |

This is the absolute value function. It is really a split function defined in

two pieces:


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The function is continuous everywhere, but only differentiable at non-zerovalues ofx.

x -2 -1 0 1 2

f(x) 2 1 0 1 2

Example 6 - Step Function

You will also encounter split functions in signal analysis (see FourierSeries and Laplace Transforms). For example, a function in electronicscan be defined as

9. Even and Odd Functions

By M. Bourne

Even Functions

A functiony =f(t) is said to be even if

f(-t) =f(t)
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for all values oft.

The graph of an even function is always symmetrical about the verticalaxis (that is, we have a mirror image through they-axis).

The waveforms shown below represent even functions:

Cosine curve

f(t) = 2 cos t

Notice that we have a mirror image through thef(t) axis.

Even Square wave:

Triangular wave:


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In each case, we have a mirror image through thef(t) axis. Another way ofsaying this is that we have symmetry about the vertical axis.

Odd Functions

A functiony =f(t) is said to be odd if

f(-t) = -f(t)

for all values oft.

The graph of an odd function is always symmetrical about the origin.

Origin Symmetry

A graph has origin symmetry if we can fold it along the vertical axis, thenalong the horizontal axis, and it lays the graph onto itself.

Another way of thinking about this is that the graph does exaclty theopposite thing on each side of the origin. If the graph is going up to theright on one side of the origin, then it will be going down to the left by thesame amount on the other side of the origin.

Examples of Odd Functions

The waveforms shown below represent odd functions.

Sine Curve

y(x) = sinx


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Notice that if we fold the curve along they-axis, then along the t-axis, thegraph maps onto itself. It has origin symmetry.

"Saw tooth" wave

Odd Square wave

Each of these three curves is an odd function, and the graphdemonstrates symmetry about the origin.


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Exercise

Sketch each function and then determine whether each function is odd oreven:

(a)

Answer

We can see from the graph that it is even.

OR: The function is even sincef(t) =f(t) for all values oft.

(b)

andf(t) =f(t+ 2)

(This last line means: Periodic withperiod = 2)

Answer
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(c)

Answer

We can see from the graph that the function is odd.

OR: The function is odd sincef(t) = -f(t) for all values oft.

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(d)

Answer

We can see from the graph that it is neitherodd nor even.

(e)

Answer
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(f)

Answer

We can see from the graph that the function is odd.

OR: The function is odd sincef(t) = -f(t) for all values oft.

Functions and graphs Problem Solver


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Below is a solver that lets you input all sorts of algebra problems and it willsolve them for you, and you can even see the steps!

The version below will show you the final answer only. You'll see a button"View steps" and this takes you to the developer's site where you can

purchase the full version of the solver (where you can see the steps).Instructions

1. Enter your problem statement (as algebra, not words!). There areexamples you can follow.

2. Choose the operation you want the solver to do (it suggestsappropriate operations, depending on your question),

3. Then click "Answer".

4. Clicking on "View steps" will lead you to a free 7-day trial account ofBagatrix.

If you are not sure how to enter your question, go to this help page (opensin a new window).

Disclaimer:IntMath.com does not guarantee the accuracy of results. Only use thissolver to check your own work.

Go to web page http://www.intmath.com/help/problem-solver.php?title=functions-and-graphs&fid=36
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functions and graphs intmath

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