Find Inverse Function - Tutorial

Examples on how to find inverse functions analytically are presented. Detailed solutions and matched exercises with answers at the end of this page are also included.

Examples with Detailed Solutions

Example 1

Find the inverse function of the linear function f given by
f(x) = 2x + 3
Solution to example 1
write the function as an equation.
y = 2x + 3
solve for x.
x = (y - 3)/2
now write f-1(y) as follows .
f
-1(y) = (y - 3)/2
or
f
-1(x) = (x - 3)/2
Check
f(f
-1(x)) = 2(f -1(x)) + 3
= 2 ((x - 3)/2) + 3
= (x - 3) + 3
= x
f
-1(f(x)) = f -1(2x+3)
= ((2x + 3) - 3)/2
= 2 x / 2
= x
conclusion: The inverse of function f given above is
f
-1(x) = (x - 3)/2
Matched Exercise 1
Find the inverse function of f given by

f(x) = - x - 4

Example 2

Find the inverse function of the quadratic function f given by
f(x) = (x - 3)2, if x ≥ 3
Solution to example 2
write the function as an equation.
y = (x - 3)
2
solve for x, two solutions .
x = 3 + √y
x = 3 - √y
the first solution is selected since x ≥ 3, write f-1(y) as follows.
f
-1(y) = 3 + √y
or
f
-1(x) = 3 + √x
Check
f(f
-1(x))=((3 + √x) - 3)2
= (√x)
2
= x
f
-1(f(x)) = 3 + √((x - 3)2)
simplifies to
= 3 + |x - 3|       ( because x ≥ 3 implies x - 3 ≥ 0 which implies |x - 3| = x - 3)
= 3 + (x - 3)
= x
conclusion
The inverse of function f given above is f
-1(x) = 3 + √x
Matched Exercise 2
Find the inverse function of f given by
f(x) = (x + 1)2, if x ≥ -1

Example 3

Find the inverse function of the rational function f given by
f(x) = (x + 1)/(x - 2)
Solution to example 3
Write the function as an equation.
y = (x + 1) / (x - 2)
Multiply both sides of the above equation by x - 2 and simplify.
y (x - 2) = x + 1
Multiply and group.
y x - 2 y = x + 1
y x - x = 2 y + 1
Factor x on the left side and solve
x (y - 1) = 1 + 2 y
x = (1 + 2 y) / (y - 1)
Change x to y and y to x
y = (1 + 2 x) / (x - 1)
The inverse of function f given above is
f
-1(x) = (1 + 2 x) / (x - 1)
Matched Exercise 3
Find the inverse function of f given by
f(x) = (x + 1)/(x - 1

Example 4

Find the inverse function of the logarithmic function f given by
f(x) = ln(x + 2) - 3
Solution to example 4
Write the function as an equation.
y = ln(x + 2) - 3
Rewrite the equation so that it is easily solved for x.
ln(x + 2) = y + 3
Rewrite the above in exponential form.
x + 2 = e
y + 3
Solve for x.
x = e
y + 3 - 2
now write f-1(y) as follows .
f
-1(y) = ey + 3 - 2
or change x into y and y into x in the above to have
f
-1(x) = ex + 3 - 2
Check
f(f
-1(x)) = f(x) = ln(ex + 3 - 2 + 2) - 3
= ln(e
x + 3) - 3
= (x + 3) - 3
= x
f
-1(f(x))=f -1(2x + 3)
= e
(ln(x + 2) - 3) + 3 - 2
= e
ln(x + 2) - 2
= x + 2 - 2
= x
conclusion: The inverse of function f given above is f
-1(x) = ex + 3 - 2
Matched Exercise 4
Find the inverse function of f given by
f(x) = 2 ln(x + 4) - 4

Answer to Matched Exercises

Answer to Matched Exercise 1
f
-1(x) = - x - 4
Answer to Matched Exercise 2
f
-1(x) = - 1 + √x
Answer to Matched Exercise 2
f
-1(x) = (x + 1)/(x - 1)
Answer to Matched Exercise 4
f
-1(x) = ex / 2 + 2 - 4

More References and Links to Inverse Functions

Find the Inverse Functions - Calculator
Applications and Use of the Inverse Functions
Find the Inverse Function - Questions
Inverse of Quadratic Functions.
Definition of the Inverse Function - Interactive Tutorial
Find Inverse Of Cube Root Functions.
Find Inverse Of Square Root Functions.
Find Inverse Of Logarithmic Functions.
Find Inverse Of Exponential Functions.
Step by Step Solver Calculator to Find the Inverse of a Linear Function.
Find the Inverse of a Square Root Function.
Find the Inverse of a Cubic Function.
Find the Inverse of a Rational Function.