Inverse of invertible functions

The inverse function of ff is denoted by f1f^{-1}.

Not to be confused with reciprocal functions f(x)1f(x)^{-1} in HL.

When we are solving for xx in an equation, we are really finding the inverse function.

Contents

Properties

The defining property of an inverse function:

f(f1(x))=f1(f(x))=xf(f^{-1}(x)) = f^{-1}(f(x)) = x

An invertible function ff satisfies

f(a)=f(b)    a=bf(a) = f(b) \iff a = b

The domain of f1f^{-1} is the range of ff; the range of f1f^{-1} is the domain of ff. You can use this without finding the inverse.

The inverse function is the mirror image of the invertible function across the diagonal line y=xy = x.

An invertible function and its inverse can only meet on the y=xy = x diagonal line.

The inverse is also invertible, such that

(f1)1=f(f^{-1})^{-1} = f

Calculating the inverse function

Remember that inverse function “undoes” the effect of the original function. That is, solve for xx in y=f(x)y = f(x). You can either solve for xx then switch xx and yy, or you can switch first then solve for yy; it is personal preference which order you do it in.

Example: Find f1(x)f^{-1}(x) given that

f(x)=(3x+1)(x1)2x2f(x) = \frac{(3x+1)(x-1)}{2x-2}

It can be shown that

y=f(x)=3x+12,    x1y = f(x) = \frac{3x+1}2, \;\; x\neq1

which is much easier to work with. Let’s switch variables and solve for yy.

x=3y+12,    y12x=3y+12x1=3y2x13=y\begin{align*} x &= \frac{3y+1}2, \;\; y\neq1 \\ 2x &= 3y + 1 \\ 2x - 1 &= 3y \\ \frac{2x-1}3 &= y \end{align*}

Note that in the original function, the point (1,2)(1, 2) is not part of the graph, so (2,1)(2, 1) is not part of the graph of f1f^{-1}.

Our final answer is

f1(x)=2x13,    x2f^{-1}(x) = \frac{2x-1}3, \;\; x\neq2 \qed
Linear function with a hole and its inverse are reflections across y = x diagonal line
f(x) and its inverse

In SL, when ask to find the inverse function, the given function will be invertible. This is not necessarily the case in HL. See next page for additional steps HL candidates should do.

Calculator

Graph inverse relations on TI-84 Plus and this is not inverse functions on TI-84 Plus