Chapter 2: Functions and Graphs Please review this lecture (from MATH 1100 class) before you begin...

Chapter 2: Functions and Graphs

Please review this lecture (from MATH 1100 class) before you begin the section 5.7 (Inverse Trigonometric functions)

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1

2.7 Inverse Functions

2.6 Combinations of Functions; Composite Functions


• Find the domain of a function.• Combine functions using the algebra of functions,

specifying domains.• Form composite functions.• Determine domains for composite functions.• Write functions as compositions.

Objectives:


Finding a Function’s Domain

If a function f does not model data or verbal conditions, its domain is the largest set of real numbers for which the value of f(x) is a real number. Exclude from a function’s domain real numbers that cause division by zero and real numbers that result in a square root of a negative number.


Example: Finding the Domain of a Function

Find the domain of the function

Because division by 0 is undefined, we must exclude from the domain the values of x that cause the denominator to equal zero.

We exclude 7 and – 7 from the domain of g.

The domain of g is

2

5( )

49x

g xx

2 49 0x 2 49x

49x

7x ( , 7) ( 7,7) (7, )


The Algebra of Functions: Sum, Difference, Product, and Quotient of Functions

Let f and g be two functions. The sum f + g, the difference, f – g, the product fg, and the quotient

are functions whose domains are the set of all real numbers common to the domains of f and

defined as follows:

1. Sum:

2. Difference:

3. Product:

4. Quotient:

fg

( ),f gg D D

( )( ) ( ) ( )f g x f x g x ( )( ) ( ) ( )f g x f x g x

( )( ) ( ) ( )fg x f x g x ( )

( )( )

f f xx

g g x


Example: Combining Functions

Let and Find each of the following:

a.

b. The domain of

The domain of f(x) has no restrictions.

The domain of g(x) has no restrictions.

The domain of is

( ) 5f x x 2( ) 1.g x x

( )( )f g x 2( 5) ( 1)x x 2 6x x

( )( )f g x

( )( )f g x ( , )


The Composition of Functions

The composition of the function f with g is denoted

and is defined by the equation

The domain of the composite function is the set of all x such that

1. x is in the domain of g and

2. g(x) is in the domain of f.

f g

( )( ) ( ( ))f g x f g x

f g


Example: Forming Composite Functions

Given and find

f g( ) 5 6f x x 2( ) 2 1,g x x x

( ( ))f g f g x 2(2 1)f x x 25(2 1) 6x x

210 5 5 6x x 210 5 1x x


Excluding Values from the Domain of

The following values must be excluded from the

input x:

If x is not in the domain of g, it must not be in the domain of

Any x for which g(x) is not in the domain of f must not be in the domain of

( )( ) ( ( ))f g x f g x

.f g

.f g


Example: Forming a Composite Function and Finding Its Domain

Given and

Find

4( )

2f x

x

1

( )g xx

( )( )f g x

( )( ) ( ( ))f g x f g x1

fx

41

2x

41

2

xx

x

41 2

xx

( )( ) ( ( ))f g x f g x


Example: Forming a Composite Function and Finding Its Domain

Given and

Find the domain of

For g(x),

For

The domain of is

4( )

2f x

x

1

( )g xx

( )( )f g x

4( )( ) ,

1 2x

f g xx

0x

12

x

( )( )f g x 1 1, ,0 0,

2 2


Example: Writing a Function as a Composition

Express h(x) as a composition of two functions:

If and then

2( ) 5h x x

( )f x x 2( ) 5,g x x ( ) ( )( )h x f g x


• Verify inverse functions.• Find the inverse of a function.• Use the horizontal line test to determine if a function

has an inverse function.• Use the graph of a one-to-one function to graph its

inverse function.• Find the inverse of a function and graph both functions

on the same axes.

Objectives:


Definition of the Inverse of a Function

Let f and g be two functions such that

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

and

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

The function g is the inverse of the function f and is denoted f –1 (read “f-inverse). Thus, f(f –1 (x)) = x and

f –1(f(x))=x. The domain of f is equal to the range of

f –1, and vice versa.


Example: Verifying Inverse Functions

Show that each function is the inverse of the other:

and

verifies that f and g are inverse functions.

( ) 4 7f x x 7( )

4x

g x

7( ( ))

4x

f g x f

7

4 74

x

7 7x x

( ( )) (4 7)g f x g x 4 7 74

x 44x x

( ( )) ( ( ))f g x g f x x


Finding the Inverse of a Function

The equation for the inverse of a function f can be found as follows:

1. Replace f(x) with y in the equation for f(x).

2. Interchange x and y.

3. Solve for y. If this equation does not define y as a function of x, the function f does not have an inverse function and this procedure ends. If this equation does define y as a function of x, the function f has an inverse function.


Finding the Inverse of a Function (continued)

The equation for the inverse of a function f can be found as follows:

4. If f has an inverse function, replace y in step 3 by

f –1(x). We can verify our result by showing that

f(f –1 (x)) = x and f –1 (f(x)) = x


Example: Finding the Inverse of a Function

Find the inverse of

Step 1 Replace f(x) with y:

Step 2 Interchange x and y:

Step 3 Solve for y:

Step 4 Replace y with f –1 (x):

( ) 2 7f x x 2 7y x 2 7x y

2 7x y 7 2x y

72

xy

1 7( )

2x

f x


The Horizontal Line Test for Inverse Functions

A function f has an inverse that is a function, f –1, if there is no horizontal line that intersects the graph of the function f at more than one point.


Example: Applying the Horizontal Line Test

Which of the following graphs represent functions that have inverse functions?

a. b.

Graph b represents a function that has an inverse.

-5 -4 -3 -2 -1 1 2 3 4 5

-5

-4

-3

-2

-1

1

2

3

4

5

x

y

-5 -4 -3 -2 -1 1 2 3 4 5

-5

-4

-3

-2

-1

1

2

3

4

5

x

y


Graphs of f and f – 1

The graph of f –1 is a reflection of the graph of f about the line y = x.


Example: Graphing the Inverse Function

Use the graph of f to draw the graph of f –1

( )y f x 1( )y f x


Example: Graphing the Inverse Function (continued)

We verify our solution by observing the reflection of the graph about the line y = x.

( )y f x1( )y f x

y x

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Chapter 2: Functions and Graphs Please review this lecture (from MATH 1100 class) before you begin...

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