Composite and inverse functions

2) Inverse functions

An inverse function is a second function which undoes the work of the first one. In this unit we describe two methods for finding inverse functions, and we also explain that the domain of a function may need to be restricted before an inverse function can exist.


Question 1

If f(x) = 7x - 2, find f^{-1}(x)

First, rearrange in terms of:
y = 7x - 2
7x - 2 = y
7x = y + 2
x = \frac{y + 2}{7}

Remember to change y back to x when you’re writing your answer:
x = \frac{y + 2}{7}, therefore f^{-1}(x)=\frac{x + 2}{7}

Question 2

Find the inverse of f(x) = \frac{2x-3}{7} .

To find the inverse of f(x), we swap the x and y values which represent the inputs and outputs. To find the inverse function we now make y the subject.

y = \frac{2x-3}{7}
x = \frac{2y-3}{7}
7x + 3 = 2y
y = \frac{7x-3}{2}

Therefore, f^{-1}(x)=\frac{7x-3}{2}

Now use the same method to solve this question.

(a) f(x) =\frac{3}{4x+4} for x > 1
(b) f(x) =\frac{x + 1}{x + 2} for x > -2

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