INVERSE FUNCTIONS

Prove that and are inverses of each other

Complete warm up individually and then compare to a neighbor

END IN MIND

Function: a relation in which each input x has exactly 1 output y

Inverse of a Function: The inverse function is a function that undoes another function: If an input x in the function ƒ produces an output y, then putting y into the inverse function g produces the output x, and vice versa.

One to One: A function is one to one if every output y, has exactly 1 input x.

Horizontal Line Test - A function f is one to one if and only if each horizontal line intersects the graph at most once.

Composition of Functions : is the process of combining two functions where one function is performed fi rst and the result of which is substituted in place of each x in the other function. The composition of functions f and g is written as f o g. [f o g](x) = f[g(x)]

VOCABULARY

INVERSE FUNCTIONS

An inverse function undoes what the function does

)(1 xf

)(1 xf

domain range

f(x)

Can you mentally determine the inverse of the functions?

SOLVING FOR AN INVERSE ALGEBRAICALLY

Finding the inverse of a function

Problem:

Replace f(x) with a y

Switch the x and y

Solve for y

Replace y with f -1(x)

23)( xxf

3

2)(1

x

xf

23 xy

23 yxyx 32

yx

3

2

FIND EACH INVERSE THEN CHECK YOUR SOLUTION WITH A

FRIEND

To find a composition of 2 functions, substitute one function for the other function:

Example: f(x)= 3x-8 and g(x) = x2+1

To find f(g(x))=f○g(x) substitute the g(x) function for the f(x) function

f(g(x))=f○g(x) = 3(x2+1)-8 =3x2+3-8 =3x2-5

FINDING COMPOSITIONS

To find g(f(x))=g○f(x), substitute the f(x) function for the g(x) function:

g(f(x))=g○f(x) = (3x-8)2+1 =(3x-8)(3x-

8)+1 =9x2-48x+64+1

=9x2-48x+65

VERIFYING INVERSES AN APPLICATION OF COMPOSITIONS

xxff

xxff

)(

)(1

1

x

xx

xxx

3

3

3

223

2223

23

To verify that two functions are inverses then,

23)( xxfUsing our earlier problem, Verify that and are inverses of each other.

3

2)(1

x

xf

Prove that and are inverses of each other

END IN MIND

Determine the inverse values of the function from the table:

END IN MIND

x -1 0 1 2

y 1 3 5 7

INVERSES GRAPHICALLY

Graphing inverse functions

The graph of the inverse of f is the reflection of f over the line y=x

31

3

2)(

2)(

xxf

xxf

Existence of an Inverse functiona function f has an inverse function if and only if

the function is one to one.One to Onea function f is one to one if for every y there is

exactly one x value

Horizontal line test

HORIZONTAL LINE TEST

a. {(4,3),(2,-1),(5,6)}

b. {(9,0),(8,1)(,4,0)}

c.

d.

DETERMINE WHETHER THE FUNCTION IS INVERTIBLE. IF IT IS, FIND IT’S INVERSE

x -5 0 7 15

y 3 6 11 15

x -1 0 1 2

y 3 3 3 3

Yes. {(3,4),(-1,2),(6,5)}

Not invertible. Since 2 y values are the same.

x 3 6 11 15

y -5 0 7 15

Not invertible since all y values are the same.

Determine the inverse values of the function from the table:

END IN MIND

x -1 0 1 2

y 1 3 5 7

x 1 3 5 7

y -1 0 1 2

USE Y=X2+5

Is the relation a function?

Graph the function.

Does the inverse exist?

How could you limit the domain so that the function will have an inverse?

Graph the inverse with the restricted domain. How can you verify that the graph of the inverse exits?

End in Mind: limit the domain so that the inverse is a function

TICKET OUT

http://www.regentsprep.org/Regents/math/algtrig/ATP8/indexATP8.htmGO to the above website for further explanations. You

must do the practice problems. Each problem will tell you if you are right or wrong. If you need help, click the explanation button