**Precalculus 1.5 Inverse Functions:**

**I. Inverse Functions**:

**A.**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*".

Under these conditions, the function "

*g*" is the

**INVERSE**of the function "

*f*":

The function "

*g*" is denoted by

*f*

^{-1}(read

*f*-inverse), so:

*f*(

*f*

^{-1}(

*x*)) =

*x*

**AND**

*f*

^{-1}(

*f*(

*x*)) =

*x*

The domain of "

*f*" must be equal to the range of

*f*

^{-1}

AND

AND

the range of "

*f*" must be equal to the domain of

*f*

^{-1}

**Example:**

Given

*f*(

*x*) = {(1,2), (4,5), (6,7)}

then

*f*

^{-1}(

*x*) = {(2,1), (5,4), (7,6)}

The graphs of "

*f*" and "

*f*

^{-1}are related to each other by the fact the graph of "

*f*

^{-1}" is a reflection of the graph of "

*f*" in the line

*y*=

*x*.

**Example:**

If the function

*has values: {(1,0), (2,3), (4,7)}*

**f**then

**f**^{-1}has values of {(0,1), (3,2), (7,4)}

**Example:**

Find the inverse of

*f*(

*x*) = 4

*x*+ 8

**1.**let

*f*(

*x*) =

*y*, so by substitution we get

*y*= 4

*x*+ 8

**2.**Switch the

*x*and

*y*

*x*= 4

*y*+ 8

3. Solve for

*y*

*x*- 8 = 4

*y*

(1/4)

*x*- 2 =

*y*

4. Now

*y*=

*f*

^{-1}so

*f*

^{-1}= (1/4)

*x*- 2

**Example:**Show that

*f*(

*x*) = (

*x*+ 8)/3 and g(

*x*) = 3

*x*- 8 are inverse functions of each other. Recall that

*f*(

*g*(

*x*)) =

*x*and

*g*(

*f*(

*x*)) = x therefore:

*f*(

*g*(

*x*)) = ( (3

*x*-8)+8)/ 3 = (3

*x*)/3 =

*x*so f(g(

*x*)) =

*x*

*g*(

*f*(

*x*)) = 3((

*x*+ 8)/3) - 8 =

*x*+ 8 - 8 =

*x*so g (

*f*(

*x*)) =

*x*

so these two functions are inverse functions of each other.

**Example:**Find the inverse function of f (x) = x

^{3}+5

1. y = x

^{3}+ 5

2. x = y

^{3}+ 5

3. x - 5 = y

^{3}

(x - 5)

^{1/3}= y

so

f

^{-1}(x) = (x - 5)

^{1/3}

**Check answer:**

f (f

^{-1}(x)) = x

((x - 5)

^{1/3})

^{3}+ 5 = x - 5 + 5 = x so it checks

**II. One - to - one function:**

A function is one-to-one if, for "a" and "b" in the domain

f (a) = f (b) implies that a = b.

A function "f " has an inverse function " f " if and only if " f " is one-to-one.

**Use the horizontal line test to check if a function is one-to-one.**If the equation can have a horizontal line pass through it only once at any value, then the function is one-to-one.

**Example**: y

^{2}= x,

doing the vertical line test (to see if it is a function) you see that when x = 1, y = -1 or y = 1 so

this is not a function

but by using the horizontal line test, the line never hits twice so therefore this equation has a one-to-one relationship.

**Example**: y = x

^{3}

this is a function and it has a one-to-one relationship so therefore

this is a one-to-one function.

Testing for One-to-one Functions:

If the function f (x) = x

^{3}+ 7

Show that f (a) = f (b)

a

^{3}+ 7 = b

^{3}+ 7

a

^{3}= b

^{3}

a = b so yes it is.

**Example:**

g (x) = 5x

^{2}+ 8

5 (a)

^{2}+ 8 = 5 (b)

^{2}+8

5a

^{2}= 5b

^{2}

a

^{2}= b

^{2}

so you could get:

a = b or

-a = b or

a = -b or

-a = -b

so therefore this function is not one-to-one.