# Find The Inverse Function from Graphs

Questions With Solutions

Find the values of the inverse of a function from its graph; examples and questions are presented along with their detailed solutions and explanations.

## Relationship Between a Function and its Inverse

If *f* is a function whose inverse is *f ^{-1}*, then the relationship between

*f*and

*f*is written as:

^{-1}*f(a) = b*⇔

*a = f*

^{ -1}(b)

## Examples

Use the graph of *f* shown below to find the following,if possible,:

a) *f ^{ -1}(5)* , b)

*f*, c)

^{ -1}(0)*f*, d)

^{ -1}(- 3)*f*, e)

^{ -1}( - 4)*f*

^{ -1}(- 5)

## Solutions

a)

According to the the definition of the inverse function

*a = f*⇔

^{ -1}(5)*5 = f(a)*

Meaning that

*a*is the value of

*x*such

*f(x) = 5*.

Using the graph below: start from y = 5 on the y-axis and draw a horizontal line to the graph of

*f*then go down to the x axis to find

*x = 3*. Therefore f(3) = 5. Hence a = 3 and therefore

*f*b)

^{ -1}(5) = 3a =

*f*⇔

^{ -1}(0)*f(a) = 0*

According to the graph shown,

*f(2) = 0*and therefore

*f*.

^{ -1}(0) = 2c)

a =

*f*

^{ -1}(- 3) ⇔ f(a) = - 3The value of x for which

*f(x) = - 3*is equal to 1 and therefore

*f*

^{ -1}(- 3) = 1d)

a =

*f*⇔

^{ -1}(- 4)*f(a) = - 4*

The value of x for which

*f(x) = - 4*is 0 and therefore

*f*.

^{ -1}(- 4) = 0e)

a =

*f*⇔

^{ -1}(- 5)*f(a) = - 5*

According to the graph of

*f*, there is no value of x for which

*f(x) = - 5*and therefore

*f*is undefined.

^{ -1}(- 5)## More Questions with Solutions

__Question 1__ - Use the graph of function *g* shown below to find the following if possible:

a) *g ^{ -1}(6) * , b)

*g*, c)

^{ -1}(0)*g*, d)

^{ -1}(- 2)*g*, e)

^{ -1}(4)*g*

^{ -1}(8)
__Question 2__ - Use the graph of function *h* shown below to find the following if possible:

a) *h ^{ -1}(1) * , b)

*h*, c)

^{ -1}(0)*h*, d)

^{ -1}(- 1)*h*

^{ -1}(2).

## Solutions to the Above Questions

__Solution to Question 1__

a) According to the the definition of the inverse function:

*a = g ^{ -1}(6)* ⇔

*6 = g(a)*

Meaning that

*a*is the value of

*x*such

*g(x) = 6*.

Using the graph below for

*x = 2, g(x) = 6*. Hence a = 2 and therefore

*g*

^{ -1}(6) = 2.

b)

*a = g*⇔

^{ -1}(0)*g(a) = 0*

According to the graph shown,

*g(- 1) = 0*and therefore

*g*.

^{ -1}(0) = - 1c) a =

*g*⇔

^{ -1}(- 2)*g(a) = - 2*

The value of x for which

*g(x) = - 2*is equal to - 2 and therefore

*g*

^{ -1}(- 2) = - 2d) a =

*g*⇔

^{ -1}(4)*g(a) = 4*

The value of x for which

*g(x) = 4*is 1 and therefore

*g*.

^{ -1}(4) = 1e) a =

*g*⇔

^{ -1}(8)*g(a) = 8*

According to the graph of

*g*, there is no value of x for which

*g(x) = 8*and therefore

*g*is undefined.

^{ -1}(8)
__Solution to Question 2__

a) According to the the definition of the inverse function:

*a = h ^{ -1}(1)* ⇔

*1 = h(a)*,

Meaning that

*a*is the value of

*x*such

*h(x) = 1*.

According to the graph shown,

*h(0) = 1*and therefore

*h*.

^{ -1}(1) = 0b) a =

*h*⇔

^{ -1}(0)*h(a) = 0*

According to the graph shown,

*h(π/2) = 0*and therefore

*h*.

^{ -1}(0) = π/2c) a =

*h*⇔

^{ -1}(-1)*h(a) = -1*

According to the graph shown,

*h(π) = - 1*and therefore

*h*.

^{ -1}(-1) = πd) a =

*h*⇔

^{ -1}(2)*h(a) = 2*

According to the graph shown, there is no value of

*x*for which

*h(x) = 2*and therefore

*h*is undefined.

^{ -1}(2)### More References and links

Find Inverse Function from Table

Inverse Function

Find Inverse Function (1) - Tutorial

Middle School Maths (Grades 6, 7, 8, 9) - Free Questions and Problems With Answers

High School Maths (Grades 10, 11 and 12) - Free Questions and Problems With Answers

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