Chapter 2 – Functions18 Days

2.1 Definition of a Function 2.2 Graphs of Functions 2.3 Quadratic Functions 2.4 Operations on Functions 2.5 Inverse Functions 2.6 Variation

Table of Contents

2.1 Definition of a Function

Two Days

A function f from a set D to a set E is a correspondence that assigns each element x of D to exactly one element y in E.

What does that mean?

Definition of a Function

Domain – The set D is the domain of the function. Domain is the set of all possible inputs.

Range – The set E is the range of the function. Range is the set of all possible outputs.

The element y in E is the value of f at x also called the image of x under f.

Domain and Range

We say that f maps D into E.

Two functions f and g are equal if and only if f(x) = g(x) for all x in D.

Function Mapping

For the function

a) Find f(a)

b) Find f(a-1)

c) Find

d) Find

Functions*

x

xxf

5)(

)(

1

xf

x

f1

Assuming that f is a linear function, find f(x) given:

Determining functions.

5)2( and 1)2( ff

p 148 (# 5,8,14-28 even, 45,47,49,52,57)

Homework

Day 2 Def Function, Domain, Range, Increasing/Decreasing, Vert Line Test, Def Linear Function Evaluating (p148 #13)

The graph of a function f is the graph of the equation y = f(x) for all x in the domain of f.

The vertical line test can be used to determine if a graph represents a function.◦ What does the vertical line test represent in terms

of a function mapping?

Graphs of Functions

-f is increasing whenf(a)<f(b) and a<b.

-f is decreasing whenf(b)>f(c) and b<c.

-f is constant when f(x)=f(y) for all x and y.

Increasing and Decreasing Functions

Given

Determine the domain of g. Evaluate g(-3)

Evaluate

Evaluating Functions

1

2)(

2 x

xxg

A

g1

What is the difference between sketching and graphing a function?

Why would we sketch a function as opposed to graph a function?

Sketching Functions

Sketch the following functions and determine the domain, range, and intervals of decreasing, increasing, and constant value:

Sketching Functions

4)( 2 xxf 225)( xxg

We can find linear functions in the same way that we find the equation of a line.

If f is a linear function such that f(-3)=6 and f(2)=-12, find f(x) where x is any real number.

Finding Linear Functions

Pg 150 #57, 59

Applications Problems

p 148 (# 15,32,34,35,46,48,50,53,54,60,63,65)

Homework

2.2 Graphs of Functions

Four Days

-Name of Family

-Parent Equation

-General Equation

-Locator Point

-Domain -Range

Parent Functions

Linear

xy

bmxy

),0(:int by

8

6

4

2

-2

-4

-6

-8

-15 -10 -5 5 10 15

f x = x

x y-2 -2-1 -10 01 12 2

-Name of Family

-Parent Equation

-General Equation

-Locator Point

-Domain -Range

Parent Functions

Value Absolute

xy

khxay

),(: khVertex

),0[

8

6

4

2

-2

-4

-6

-8

-15 -10 -5 5 10 15

f x = x

x y-2 2-1 10 01 12 2

-Name of Family

-Parent Equation

-General Equation

-Locator Point

-Domain -Range

Parent Functions

Quadratics

2xy

khxay 2)(

),(: khVertex

),0[

8

6

4

2

-2

-4

-6

-8

-15 -10 -5 5 10 15

f x = x2

x y-2 4-1 10 01 12 4

-Name of Family

-Parent Equation

-General Equation

-Locator Point

-Domain -Range

Parent Functions

Cubics

3xy

khxay 3)(

),(: khInflection

8

6

4

2

-2

-4

-6

-8

-15 -10 -5 5 10 15

f x = x3

x y-2 -8-1 -10 01 12 8

-Name of Family

-Parent Equation

-General Equation

-Locator Point

-Domain -Range

Parent Functions

Root Square

xy

khxay

),(:Endpoint kh

),0[ ),0[

8

6

4

2

-2

-4

-6

-8

-15 -10 -5 5 10 15

f x = x

x y0 01 14 29 316 4

-Name of Family

-Parent Equation

-General Equation

-Locator Point

-Domain -Range

Parent Functions

Root Cube

3 xy

khxay 3

),(:Inflection kh

8

6

4

2

-2

-4

-6

-8

-15 -10 -5 5 10 15

f x = x1

3

x y-8 -2-1 -10 01 18 2

Parent: Shift up k units: Shift down k units:

Shift right h units: Shift left h units

Combined Shift:◦ (right h units, up k units)

Graph Shifting and Reflections

xy

kxy

kxy

hxy

hxy

khxy

)(xfy

khxfy )(

)( hxfy

)( hxfy

kxfy )(

kxfy )(

Parent: Reflection in x-axis:

Vertical Stretch a>1 Vertical Shrink 0<a<1

Horizontal Stretch 0<c<1 : Horizontal Compression c>1:

Combined Transformation:

Graph Shifting and Reflections

xy

xay

xcy

xay

)(xfy

)(xfay

)( xcfy

)(xfay

khxay khxfay )(

Graph the following using translations:

Graph Shifting and Reflections

12 xy

33 xy

31 2 xy

542 xxy

33 xy

83 xy

242 xy

22xy

142 2 xy

32 xy

Shifts and Reflections WS

Homework

Day 2 – Even and Odd functions. Vertical and Horizontal stretching and compressing of graphs.

f is an even function if f(-x)=f(x) for all x in the domain.◦ Even functions have symmetry with respect to the

y-axis.◦ Ex:

f is an odd function if f(-x)=-f(x) for all x in the domain. ◦ Odd functions have symmetry with respect to the

origin. ◦ Ex:

Even and Odd Functions

2)( xxf

3)( xxf

A parent function is the simplest function in a family of certain characteristics.

A translation shifts the graph horizontally, vertically, or both. Resulting in a graph of the same shape in a different location.

A reflection over the x-axis changes y-values to their opposites.

Family Functions and Shifts

A vertical stretch multiplies all y-values by the same factor greater than 1.

A vertical shrink reduces all y-values by the same factor between 0 and 1.

Each member of a family of functions is a transformation, or change, of the parent function.

A horizontal compression divides all x-values by the same factor greater than 1.

A horizontal stretch divides all x-values by the same factor between 0 and 1.

Family Functions and Shifts

Parent: Shift up k units: Shift down k units:

Shift right h units: Shift left h units

Combined Shift:◦ (right h units, up k units)

Graph Shifting and Reflections

xy

kxy

kxy

hxy

hxy

khxy

)(xfy

khxfy )(

)( hxfy

)( hxfy

kxfy )(

kxfy )(

Parent: Reflection in x-axis:

Vertical Stretch a>1 Vertical Shrink 0<a<1

Horizontal Stretch 0<c<1 : Horizontal Compression c>1:

Combined Transformation:

Graph Shifting and Reflections

xy

xay

xcy

xay

)(xfy

)(xfay

)( xcfy

)(xfay

khxay khxfay )(

pg 164 (# 2,3,5,7,8,13,15,17,20,31-36,39 a-f, 41,42,45)

Homework

Day 3 – Piecewise functions and questions from the previous 2 days. Application of Piecewise functions (pg 168 #66)

Piecewise functions are defined by more than one expression over different intervals.

Absolute Value is actually a piecewise defined function.

Piecewise Functions

Lets graph the following piecewise defined function.

Piecewise Functions

2x;

2x2;

2x;

32

4

)( 2

x

x

x

xf

Lets graph the following piecewise defined function.

Piecewise Functions

2x;

2x2;

2x;

2

42

)(2

x

x

x

xf

An electric company charges its customers $0.0577 per kWh for the first 1000kWh, $0.0532 for the next 4000kWh, and $0.0511 for any over 5000kWh. Write a piecewise defined function C for a customer’s bill of x kWhs.

How much will a customer’s bill be if they used 4300kWh of electricity?

Applications of Piecewise Functions

pg 167 (# 47-50,53,54,55,56,63-65)

Homework

Day 4 – Graphing Piecewise functions WS. Working day for students.

Graphing Piecewise Functions WS

Homework

2.3 Quadratic Functions

Two Days

Day 1 – Standard form of a quadratic. Vertex form of a quadratic. Completing the square. Finding x and y intercepts.

Standard form of a Quadratic:

Vertex form of a Quadratic:

Quadratic Functions

0 ; 2 acbxaxy

),( :Vertex ; )( 2 khkhxay

Competing the Square

463 2 xxy

4) 2(3 2 xxy222 )1(34))1( 2(3 xxy

3a (-1,-7),:Vertex 7)1(3 2 xy

● To find the x-intercept, set y=0. Solve for x.

● To find the y-intercept, set x=0. Solve for y.

Find the x and y intercepts of the following:

Finding x and y intercepts

14 xy 652 xxy

Vertex and Intercepts WS

Homework

Day 2 – Vertex formula and Theorem on Max/Min Values.

Standard Form of a Quadratic

Factored Form of a Quadratic

Vertex Formula

a

bf

a

bcbxaxy

2,

2 :Vertex ; 2

2

,2

:Vertex ; )()( 212121

xxf

xxxxxxay

Theorem on the Maximum or Minimum of a Quadratic Function

is 2

then 0,a where)( If 2

a

bfcbxaxxf

0a if of valuemaximum the)1( f

0a if of valueminimum the)2( f

Find the vertex of the following and determine if it is a max or a min:

Finding the Vertex

54 xy

1242 xxy

1052 xxy

The length of a frog’s leap is 9ft and has a maximum height of 3ft off the ground. Assuming the frog’s path through the air is parabolic, find an equation that describes the path of the frog through the air.

Applications of Quadratics

pg 179 (# 1,6,8,10,13,17,20,23,26,27,30,39)

pg 179 (# 7,15,31,38,40,43,44,49,54)

Homework

2.4 Operations on Functions

One Day

We can perform several operation on functions just as we perform the same operation on real numbers. Consider f(x) and g(x):

Operation on Functions

)()())(( xgxfxgf

)()())(( xgxfxgf

)()())(( xgxfxgf

)(

)()(

xg

xfx

g

f

The composite function of two functions f and g is defined by:

The domain of is the set of all x in the domain of g such that g(x) is in the domain of f.

Essentially, the range of g(x) is the domain of f(x) minus any possible restrictions.

Composition of Functionsgf

))(())(( xgfxgf

gf

Let and

Find:

Examples23)( xxf xxxg 2)( 2

))(( xgf

))(( xgf

))(( xgf

)(x

g

f

))(( xgf

))(( xfg

Is ????

Consider

Question..fggf

xxf )( 4)( xxg

)4)(( gf

)5(

f

g

)3)(( gf

)2)(( fg

pg 192 (# 2,3,6,7,9,12,15,21,25,29,30,35,36,39,45,46)

Homework

Graph each function belowa) Exact x and y interceptsb) Give the domain/rangec) Intervals of increase, decrease, constant

1)

2)

3)

4)

Graph Shifting Review

1)1(2 2 xy

653 xy

1)4( 3 xy

462 xy

2.5 Inverse FunctionsTwo Days

f(x) and g(x) are inverse functions that “undo” one another if and only if .

Notation:Original Function:

Inverse Function:

Inverses are NOT reciprocals!!

Definition of an Inverse Function

xxfgxgf ))(())((

)(xf

)(1 xf

Inverses switch the x and y values of a function. (x,y) -> (y,x)

The domain of f is the range of its inverse. The range of f is the domain of its inverse.

Graphically, the inverse of a function is a reflection over y=x.

Properties of Inverses

Can you come up with a function that when reflected over the line y=x will no longer be a function?

Do all Functions have Inverses?

In order for a function to have an inverse, a fucntion must be 1 to 1. That is, no two elements in the domain can have the same y value.

Ex: is not a 1-1 function.

We can however restrict the domain to find partial inverses. i.e has an inverse.

1-1 Fucntions

)4,2( and )4,2( )( 2 xxf

]0|[ )( 2 xxxxf

To find an inverse:◦ 1. Check if the function is 1-1. Restrict the

domain if the original function is not 1-1.◦ 2. Write f(x) as y.◦ 3. Switch all x and y in the equation.◦ 4. Solve for the “new” y.◦ 5. Rewrite the domain if necessary.◦ 6. Check that or graph on

the TI and check for symmetry about y=x.

Finding Inverses

xxfgxgf ))(())((

Find the inverses to the following functions:

Examples

34)( xxf

3)( xxf

1)( 2 xxf

pg 203 (# 4,5,7,9-12,14,15,19,23,29,31,35)

Homework

Finding Inverses WS (# 1-7)

Homework

2.6 VariationOne Day

A variation or proportion is used to describe relationships between variable quantities.

k is a nonzero real number called a constant of variation or constant of proportionality.

Variation

Direct Variation◦ y varies directly with x◦ y is directly proportional to x

Inverse Variation◦ y varies inversely with x◦ y is inversely proportional to x

Combined or Joint Variation◦ z varies jointly with x and y◦ z varies directly with x and inversely

with y

Types of Variation

kxy

x

ky

kxyz

y

kxz

V varies jointly as B and H

P varies directly as the square of V and inversely as R

The volume, V, of a gas varies directly as the temperature, T, and inversely as the pressure, P

The distance, D, that a free-falling object falls varies directly as the square of the time, T, that it falls

Write a Variation Equation for the Following:

1. Write the general formula that variables and a constant of variation.

2. Substitute the initial conditions for the variables and solve for the constant of variation k.

3. Substitute the constant of variation k into the general formula from your first step.

4. Use you general formula to solve the problem.

Solving Variation Problems

The price, P, of a diamond is directly proportional to the square of the weight, W. If a 1 carat diamond costs $2000, find the price of a 0.7 carat diamond.

Example

The electrical resistance R of a wire varies directly as its length l and inversely as the square of its diameter d. A wire 100ft long of diameter 0.01in has a resistance of 25ohms. Find the resistance of a wire with a diameter of 0.015in and 50ft of length.

Example

pg 209 (# 2-4,6,8,10,13,14,16,17,20,21,23)

Homework