Unit 2Functions and Their Graphs

Section ASets

Set: a collection of unique elements

By Roster

Write a roster for the set of all integers between 1 and 10- not including 1 or 10.

Set Builder Notation

Real Numbers:Integers:Natural Numbers:

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Use set builder notation to write the set of all real numbers, such that, x is less than -4.5

Interval Notation

Not included/open:Included/closed:

Write the following inequality using interval notation: 2≤x<6

How do you write a two-part inequality in interval notation?Ex: x<-1 or x>8

Write in interval notation:x<0 or 2<x≤10

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Section 1Functions

Definition of a FunctionA function f from a set A to a set B is a rule of correspondence that assigns to each element x in the set A exactly one element y in the set B. The set A is the domain (or set of inputs) of the function f, and the set B contains the range (or set of outputs).

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Example 1.1Determine if each of the following is a function or not.a. The input value x is the number of representatives from a

state, and the output value y is the number of senators.

b.

c.

y=x2

This is the most common way to represent a function

Independent Variable: Dependent Variable:

The domain is the set of all values taken on by _____ and the range is the set of all values taken on by _____.

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Example 1.2Determine if each of the following is a function of x.a. x2+y=1

b. -x+y2=1

Try This:Do these equations represent y as a function of x?a. x2+y2=8

b. y-4x2=36

Function NotationWe know that y=1-x2 gives y as a function of x. We can name this function "f" and then use function notation to describe it:

Input: Output:Equation:

So y=f(x).Remember f is the name of the function and f(x) is the value of the function at x.

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To find a function value, substitute the specified input value into the given equation.

f(x)=3-2xFind f(-1)

F and x are the most commonly used labels for functions, but other letters can be used as well.f(x)=x2-4x+7f(t)=t2-4t+7g(s)=s2-4s+7All these define the same function.

Example 1.3Let g(x)=-x2+4x+11. g(2)

2. g(t)

3. g(x+2)

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Try This:g(t)=10-3t2

a. g(2)

b. g(-4)

c. g(x-1)

Example 1.4Evaluate the function when x=-1, 0, and 1.

Example 1.5Find all real values of x such that f(x)=0.a. f(x)=-2x=10

b. f(x)=x2-5x+6

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Example 1.6Find the values of x for which f(x)=g(x).a. f(x)=x2+1 and g(x)=3x-x2

b. f(x)=x2-1 and g(x)=-x2+x+2

Domain of a FunctionSome domains are described exactly by the function, others are implied. The implied domain is the set of all real numbers for which the expression is defined.

f(x)=

has an implied domain of all real x other than x=2 and -2.

f(x)=√x

has an implied domain of x≥0

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Example 1.7Find the domain of each function.a. f: {(-3,0), (-1,4), (0,2), (2,2), (4,-1)}

b. g(x)=

c. Volume of a sphere: V=4/3πr3

d. h(x)=√4-3x

Example 1.8You work in the marketing department of a soft-drink company and are experimenting with a new soft-drink can that is slightly narrower and taller than a standard can. For your experimental can, the ratio of the height to the radius is 4.Express the volume of the can as a function of the radius r.

Express the volume of the can as a function of the height h.

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Example 1.9A baseball is hit at a point 3 feet above the ground at a velocity of 100 feet per second and an angle of 45 degrees. The path of the baseball is given by the function

y=-0.0032x2+x+3where y and x are measured in feet. Will the baseball clear a 10-foot fence located 300 feet from home plate?

Difference Quotients

Example 1.10Find the difference quotient for f(x)=x2-4x+7

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Section 2Analyzing Graphs of Functions

The graph of a function f is the collection of ordered pairs (x, f(x)) such that x is in the domain of f.

x= the directed distance from the y-axisf(x)= the directed distance from the x-axis

Example 2.1Use the graph of the function f to find (a) the domain of f, (b) the function values f(-1) and f(2), and (c) the range of f.

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Vertical Line Test for FunctionsA set of points in a coordinate plane is the graph of y as a function of x if and only if _______________________________________ ______________________________________________________

Do these graphs represent y as a function of x?

Zeros of a FunctionThe zeros of a function f of x are the x-values for which ________

Example 2.2Find the zeros of each function.a. f(x)=3x2+x-10

b. g(x)=√10-x2

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c. h(t)=

Increasing, Decreasing, and Constant FunctionsA function f is increasing on an interval if, for any x1 and x2 in the interval, _____________________________________A function f is decreasing on an interval if, for any x1 and x2 in the interval, _____________________________________A function f is constant on an interval if, for any x1 and x2 in the interval, ____________________

Example 2.3Use the graphs to describe the increasing, decreasing, or constant behavior of each function.

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Relative Minimum and Maximum ValuesA function value f(a) is called a relative minimum of f if there exists an interval (x1, x2) that contains a such that

A function value f(a) is called a relative maximum of f if there exists an interval (x1, x2) that contains a such that

Example 2.4Use a graphing calculator to find the relative minimum of the function f(x)=3x2-4x-2.

Example 2.5Find the relative minimum(s) and relative maximum(s) of the function f(x)=-x3+x.

Example 2.6During the 1980s, the average price of a 1-carat polished diamond decreased and then increased according to the model

C=-0.7t3+16.25t2-106t+388, 2≤t≤10where C is the average price in dollars and t represents the calendar year, with t=2 corresponding to January 1, 1982. According to this model, during which years was the price of diamonds decreasing?

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During which years was the price of diamonds increasing? Approximate the minimum price of a 1-carat diamond between 1982 and 1990.

Average Rate of ChangeThe average rate of change between any two points on a graph is the slope of the line through the two points. This line is called the _____________________

Example 2.7Find the average rates of change of f(x)=x3-3x froma. x=-2 to x=-1

b. x=0 to x=1

Even and Odd FunctionsA function is even if, for each x in the domain of f, ____________A function f is odd if, for each x in the domain of f, ____________

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Example 2.8a. Is the function g(x)=x3-x even or odd?

b. Is the function h(x)=x2+1 even or odd?

Section 3Parent Functions

Linear Functionf(x)=mx+b• the domain of the function is the set of ___________________

• when m≠0, the range of the function is the set of ___________ ______________________

• the graph has an x-intercept and a y-intercept

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• the graph is increasing when _________, decreasing when ___________, and continuous when ___________

Example 3.1Write the linear function f for which f(1)=3 and f(4)=0.

Special Linear Functions

Constant Function:

Identity Function:

Squaring Functionf(x)=x2

• U-shaped curve

• the domain is the set of _______________________________

• the range is the set of ________________________________

• the function is ______________

• the graph has an intercept at ___________

• the graph is decreasing on the interval ______________ and

increasing on the interval _______________

• the graph is symmetric with respect to the ______________

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• the graph has a relative minimum at ______________

Cubic Functionf(x)=x3

• the domain is the set of _______________________________

• the range is the set of ________________________________

• the function is ____________

• the graph has an intercept at _________

• the graph is increasing on the interval ____________

• the graph is symmetric with respect to the ____________

Square Root Functionf(x)=√x• the domain is the set of _______________________________

• the range is the set of ________________________________

• the graph has an intercept at _________

• the graph is increasing on the interval __________

Reciprocal Functionf(x)=1/x

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• the domain is __________________

• the range is ___________________

• the function is __________

• the graph does not have any intercepts

• the graph is decreasing on the intervals __________________

• the graph is symmetric with respect to the ____________

Step functionsFunctions whose graphs represent stairstepsMost famous is the greatest integer function:f(x)=[[x]]=the greatest integer less than or equal to x• the domain is the set of _______________________________

• the range is the set of ______________________

• the graph has a y-intercept at ___________ and x-intercepts in

the interval ____________

• the graph is ______________ between each pair of

consecutive integer values of x

• the graph jumps vertically one unit at each _______________

Example 3.2Evaluate the function when x=-1, 2, and 3/2

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f(x)=[[x]]+1

Example 3.3Sketch the graph of:

Parent Functions

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Section 4Transformations of Functions

Vertical ShiftsExample: f(x)=x2 Shifted up three units.

Example: g(x)=√x Shifted down 7 units.

Horizontal ShiftsExample: f(x)=x2 shifted right 4 units.

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Example: g(x)=|x| shifted left 2 units.

Vertical and Horizontal ShiftsLet c be a positive real number. Vertical and horizontal shifts in the graph of y=f(x) are represented as follows:

1. Vertical shift c units upward:

2. Vertical shift c units downward:

3. Horizontal shift c units to the right:

4. Horizontal shift c units to the left:

Example 4.1Describe the graph of the following functions in relationship to the graph of f(x)=x3

a. g(x)=x3+1

b. h(x)=(x-1)3

c. k(x)=(x+2)3+1

Example 4.2

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Each of the graphs shown is a vertical or horizontal shift of the graph of f(x)=x2. Find an equation for each function.

ReflectionsReflections in the coordinate axes of the graph of y=f(x) are represented as follows:

1. Reflection in the x-axis:

2. Reflection in the y-axis:

Example 4.3This is the graph of f(x)=x4

Use this to write the equation of each of these functions:

Example 4.4

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Compare the graph of f(x)=√x with each of the following:a. g(x)=-√x

b. h(x)=√-x

c. k(x)=-√x+2

Nonrigid TransformationsWhat is a nonrigid transformation?

If a nonrigid transformation on the graph of y=f(x) is represented by y= cf(x)

If c>1 it is a

If 0<c<1 it is a

If a nonrigid transformation on the graph of y=f(x) is represented by y=f(cx)

If c>1 it is a

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If 0<c<1 it is a

Example 4.5Compare the graph of each function with the graph of f(x)=|x|a. h(x)=3|x|

b. g(x)=(1/3)|x|

Example 4.6Compare the graph of each function with the graph of f(x)=2-x2

a. g(x)=2-8x3

b. h(x)=2-(1/8)x3

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Section 5Combinations of Functions and Composite Functions

Arithmetic Combinations of FunctionsJust like with real numbers, two functions can be combined by addition, subtraction, multiplication, and division to form new functions.

f(x)=2x-3 g(x)=x2-1Find the sum, difference, product, and quotient of f and g.

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Example 5.1Find (f+g)(x) for the functions

f(x)=2x+1 and g(x)=x2+2x-1Then evaluate the sum when x=3 -- is this value the same as evaluating f(3)+g(3)?

Example 5.2Find (f-g)(x) for the functions

f(x)=2x+1 and g(x)=x2+2x-1Then evaluate the difference when x=2.Is this value the same as f(2)-g(2)?

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Example 5.3Given f(x)=x2 and g(x)=x-3 find (fg)(x). Then evaluate the product when x=4.

Example 5.4Find (f/g)(x) and (g/f)(x) for the functions

f(x)=√x and g(x)=√4-x2

Then find the domains of f/g and g/f

Compositions of Functionsf(x)=x2 and g(x)=x+1The composition of f with g is:f(g(x))=f(x+1)=(x+1)2

Also denoted f∘g

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Example 5.5Given f(x)=x+2 and g(x)=4-x2, find:a. (f∘g)(x)

b. (g∘f)(x)

c. (g∘f)(-2)

Example 5.6Find the composition (f∘g)(x) for:

f(x)=x2-9 and g(x)=√9-x2

Then find the domain of (f∘g)***Note the domain of f and g individually!

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Function “Decomposition”We need to be able to identify two functions that make up a given composite function.h(x)=(3x-5)3

What two functions were combined to create h(x)?

Example 5.7Express the function

h(x)=

as a composition of two functions.

Example 5.8The number of bacteria in a refrigerated food is given by

N(T)=20T2-80T+500, 2≤T≤14where T is the Celsius temperature of the food. When the food is removed from refrigeration, the temperature is given by

T(t)=4t+2, 0≤t≤3

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where t is the time in hours. Find:a. The composite N(T(t)). What does this function represent?

b. The number of bacteria in the food when t=2 hours.

c. The time when the bacteria count reaches 2000.

Section 6Inverse Functions

InversesInverse functions have the effect of "undoing" each otherInverse of f(x) is denoted by ___________The domain of f is the _____________The range of f is the _____________The composition of f with f-1 (and that of f-1 with f) is the ________________________

For Example:If f(x)=x+4, then f-1(x)=f(f-1(x))=

f-1(f(x))=

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f(x)=x+4={(1,5), (2,6), (3,7), (4,8)}f-1(x)=x-4={(5,1), (6,2), (7,3), (8,4)}Domain of f(x):

Range of f(x):

Domain of inverse:

Range of inverse:

Example 6.1Find the inverse of f(x)=4x. Then verify that both f(f-1(x)) and f-1(f(x)) are equal to the identity function (both equal x when simplified).

Definition of InverseLet f and g be two functions such that

f(g(x))=x for every x in the domain of gand

g(f(x))=x for every x in the domain of f.Under these conditions, the function g is the inverse of the function f. The function g is denoted by f-1 (read "f-inverse"). Thus,

f(f-1(x))=xand

f-1(f(x))=x.The domain of f must be equal to the range of f-1, and the range of f must be equal to the domain of f-1.

Example 6.2Show that the functions are inverses of each other:

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Example 6.3Which of the functions is the inverse of f(x)=5/(x-2)?

The Graph of an Inverse FunctionThe graphs of a function f and its inverse function f-1 are related to each other in the following way.

If the point (a, b) lies on the graph of f, then the point (b, a) must lie on the graph of f -1 , and vice versa.

This means that the graph of f-1 is a reflection of the graph of f in the line y = x.

The Existence of an InverseConsider f(x)=x2

We would assume the inverse to be f-1(x)=_____________But, according to the rules of inverses, the domain of f must be the range of f-1. In this case, the domain of f is ___________________,

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and the range of f-1 is ____________________.To have an inverse, a function must be one-to-one.

One-to-one: For every a and b in the domain of f, f(a)=f(b) implies that ______________ This means that each x is mapped to exactly one y and each y is mapped to exactly one x.

Example 6.4Look at the graph of each function to determine whether or not the function has an inverse function.a. f(x)=x3-1 b. g(x)=x2-1

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Example 6.5Find the inverse (if it exists) of

f(x)=

Example 6.6Find the inverse of f(x)=√2x-3

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Example 6.7Find the inverse of:a. f(x)=x+4

b. g(x)=2x-1

c. h(x)=-3x+1

Example 6.8Find the inverse of each:a. f(x)=x6, x≥0

b. g(x)=-x3+4

c. h(x)=1/27 x3

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d. k(x)=2x5+3

Example 6.9Your hourly wage is $7.50 plus $0.90 for each unit x produced per hour. Let f(x) represent your weekly wage for 40 hours of work. Does this function have an inverse? If so, what does the inverse represent?

Example 6.10Let x represent the retail price of an item (in dollars), and let f(x) represent the sales tax on the item. Assume that the sales tax is 7% of the retail price AND that the sales tax is rounded to the nearest cent. Does this function have an inverse? (Hint: Can you undo this

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function? For instance, if you know that the sales tax is $0.14, can you determine EXACTLY what the retail price is?)

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