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 1Find the inverse function of the linear function f given bySolution to example 1write 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 1Find the inverse function of f given by
## Example 2Find the inverse function of the quadratic function f given by^{2}, if x ≥ 3
Solution to example 2write 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 conclusionThe inverse of function f given above is f ^{ -1}(x) = 3 + √x
Matched Exercise 2Find the inverse function of f given by ^{2}, if x ≥ -1
## Example 3Find the inverse function of the rational function f given bySolution to example 3Write 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 3Find the inverse function of f given by
## Example 4Find the inverse function of the logarithmic function f given bySolution to example 4Write 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) = e^{y + 3} - 2
or change x into y and y into x in the above to have f ^{ -1}(x) = e^{x + 3} - 2
Check f(f ^{ -1}(x)) = f(x) = ln(e^{x + 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) = e^{x + 3} - 2
Matched Exercise 4Find the inverse function of f given by ## Answer to Matched ExercisesAnswer to Matched Exercise 1f ^{ -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) = e^{x / 2 + 2} - 4
## More References and Links to Inverse FunctionsFind the Inverse Functions - CalculatorApplications 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. |