# 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

f
-1(x) = - x - 4
f
-1(x) = - 1 + √x
f
-1(x) = (x + 1)/(x - 1)
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