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 -1 is written as:

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 -1(0) ,   c) f -1(- 3) ,   d) f -1( - 4) ,   e) f -1(- 5)

function given by graph
Figure 1. Graph of a function, example 1


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 -1(5) = 3
function given by graph, solution
Figure 2. Graph of a function, example 1, solution
b)
a =
f -1(0)   ⇔    f(a) = 0
According to the graph shown,
f(2) = 0 and therefore f -1(0) = 2 .
c)
a =
f -1(- 3)   ⇔   f(a) = - 3
The value of x for which
f(x) = - 3 is equal to 1 and therefore f -1(- 3) = 1
d)
a =
f -1(- 4)   ⇔    f(a) = - 4
The value of x for which
f(x) = - 4 is 0 and therefore f -1(- 4) = 0 .
e)
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 -1(- 5) is undefined.


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 -1(0) ,   c) g -1(- 2) ,  d) g -1(4) ,  e) g -1(8)

function g given by its graph
Figure 3. Graph of function g, question 1

Question 2 - Use the graph of function h shown below to find the following if possible:
a) h -1(1) ,  b) h -1(0) ,  c) h -1(- 1) ,  d) h -1(2)

.

function h given by its graph
Figure 4. Graph of function h, question 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

function given by graph solution .


b) a = g -1(0)   ⇔    g(a) = 0
According to the graph shown,
g(- 1) = 0 and therefore g -1(0) = - 1 .
c) 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) = - 2
d) a = g -1(4)   ⇔    g(a) = 4
The value of x for which
g(x) = 4 is 1 and therefore g -1(4) = 1 .
e) 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 -1(8) is undefined.



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) = 0.
b) a = h -1(0)   ⇔   h(a) = 0
According to the graph shown, h(π/2) = 0 and therefore h -1(0) = π/2.
c) a = h -1(-1)   ⇔   h(a) = -1
According to the graph shown, h(π) = - 1 and therefore h -1(-1) = π.
d) 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 -1(2) is undefined.


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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