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.

Example 1: Find the inverse function of 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

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

=2x/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 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 + sqrt(y)

x = 3 - sqrt(y)

the first solution is selected
since x >= 3, write f^{-1}(y) as follows.

f^{ -1}(y) = 3 + sqrt(y)

or

f^{ -1}(x) = 3 + sqrt(x)

Check

f(f^{ -1}(x))=((3+sqrt(x))-3)^{2}

=(sqrt(x))^{2}

=x

f^{ -1}(f(x))=3+sqrt((x-3)^{2})

=3+|x-3| (since x >= 3, x-3 >= 0, |x-3| = x-3)

=3+(x-3)

=x

conclusion:
The inverse of function f given above is
f^{ -1}(x) = 3 + sqrt(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 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 - 2y = x + 1

y x - x = 2y + 1

Factor x on the left side and solve

x(y - 1) = 1 + 2y

x = (1 + 2y) / (y - 1)

Change x to y and y to x
y = (1 + 2x) / (x - 1)

The inverse of function f given above is
f^{ -1}(x) = (1 + 2x) / (x - 1)

Matched Exercise 3: Find the inverse function of f given by