Inverse functions

Before we move on to inverse functions, we discuss function composition. Suppose we have two functions $f(x)$ and $g(x)$; the composition of $f(x)$ and $g(x)$, denoted by $(f \circ g)(x)$, is evaluated by plugging the second function in the first function, as follows

note that the order is important and interchanging the order generally gives a different result. Consider $f(x)=x+5$ and $g(x)=x-5$, then

and

In this case, the compositions are the same, both yielding $x$,

Choosing a value for $x$ and plugging it in the composition of $f$ and $g$ (or of $g$ and $f$ ), we end up with that same value of $x$, i.e. for $x=3$,

and a similar result is obtained for $g(f(3))$. Whatever operation the first function does to the value of $x$, the second one undoes it. An inversefunction, denoted by $f^{-1}$ is a rule that undoes $f$ 's rule (e.g. addition and subtraction are inverse operations).

A function $f(x)$ is one-to-one $(1-1)$ if $f(a)=f(b)$ whenever $a=b$ i.e. for each $x$ value, there exists one unique $y$ value. For example, $f(x)=e^{x}$ is a 1 - 1 function while $f(x)=\sin x$ on $0 \leq x \leq \pi$, is not. Inverse functions exist only for $1-1$ functions; if $f$ is not $1-1$, then $f^{-1}$ maps $y$ back to more than one $x$ value, violating the definition of a function.

Definition 1.1 Given two, one-to-one functions $f(x)$ and $g(x)$, if

then $g(x)$ is the inverse of $f(x)$, denoted by $g(x)=f^{-1}(x)$ and $f(x)$ is the inverse of $g(x)$, denoted by $f(x)=g^{-1}(x)$.

For a 1 - 1 function $f$ with domain $A$ and range $B$, its inverse $f^{-1}$ has domain $B$ and range $A$.

Inverse function theorem

If $f$ is strictly increasing or decreasing, then $f$ is a 1 - 1 function and invertible. A function is increasing if $f\left(x_{1}\right) \geq f\left(x_{2}\right)$ and strictly increasing if $f\left(x_{1}\right)>f\left(x_{2}\right)$ whenever $x_{1}>x_{2}$ for all $x_{1}$ and $x_{2}$ in its domain. Similarly, a function is decreasing if $f\left(x_{1}\right) \leq f\left(x_{2}\right)$ and strictly decreasing if $f\left(x_{1}\right)<f\left(x_{2}\right)$ whenever $x_{1}>x_{2}$. An example of a strictly increasing function is $f(x)=e^{x}$ and an example of a strictly decreasing function over $\mathbb{R}$ is $f(x)=1 /\left(e^{x}+1\right)$.

Suppose that we have an increasing function $f(x)$; then, any tangent drawn to the curve of $f$ at any point is positively sloped which implies that the derivative, denoted by $f^{\prime}(x)$ (this is the rate of change of $f$ with $x$ ), is positive ${ }^{2}$. Note that this is a sufficient but

${ }^{2} \mathrm{~A}$ number is positive if it is greater than zero and negative if it is less than zero. A number is nonnegative if it is greater than or equal to zero. not necessary condition for a function to be strictly increasing. For instance, consider $f(x)=x^{3}$; this is a strictly increasing function since $f$ increases as $x$ increases. However, $f^{\prime}(x)$ is not positive at all values of $x$ since at $x=0, f^{\prime}(0)=0$. Increasing and strictly increasing functions never have a negative gradient. Further, strictly increasing functions can have a zero gradient provided that this occurs at what is referred to as an inflection point (like in the $x^{3}$ example above).

Finding inverses

Given a function $f$ (assumed to be $1-1$ ), we want to find the inverse function, $f^{-1}$. We outline the steps required to construct $f^{-1}$ from $f$ next.

For a function $y=f(x)$, replace every $x$ with $y$ and every $y$ with $x$. Next, solve the resulting equation for $y$. This is the inverse function since $x=f^{-1}(y)$. To verify, check that $\left(f \circ f^{-1}\right)(x)=\left(f^{-1} \circ f\right)(x)=x$.

Example 1.3 Given $f(x)=x / 2+1$, find $f^{-1}(x)$

Solution We write $y=x / 2+1$ and replace all $x$ 's with $y$ and $y$ 's with $x$ to give $x=y / 2+1$. Solving for $y$,

Replacing $y$ with $f^{-1}$ gives the inverse. To verify the result,

and

There is an interesting relationship between the graphs of the functions $f$ and $f^{-1}$. Given a function $y=f(x)$, to plot the graph of the inverse, we swap the $(x, y)$ coordinates of every point on the graph. Geometrically, this is represented by reflecting the graph of $f$ about the line $y=x$ as shown in Fig. 1.2; this is always the case with the graphs of a function and its inverse.

Derivatives

Derivatives are discussed in detail in Chapter 3; however for the purposes of this section we use the definition of the derivative as a slope of a curve at a point or, simply, the slope of the tangent line.

The derivatives of the function and its inverse are related; if $(a, b)$ lies on the graph of $f$ then $(b, a)$ lies on the graph of $f^{-1}$ (as shown in Fig. 1.2). The slopes of the tangent lines at the inverse points are reciprocals of each other; this is true for all slopes of inverses. The graph of function $f$ is $y=f(x)$ or equivalently $x=f^{-1}(y)$ so, we have

Figure 1.2: The graph of the inverse of a function $f(x)$ is a reflection of $f$ about the line $y=x$. For every point $(x, y)$ on the graph of $f$, there is a corresponding point $(y, x)$ on the graph of the inverse function.

Let $f$ be a 1 - 1 function that is differentiable on an interval $\mathcal{I}$. If $f^{-1}$ is the inverse of $f$ and $f^{\prime}(x) \neq 0$ then,

Note that the above result is a consequence of taking the derivative with respect to (wrt) $x$ on both sides of

As an example, consider $f(x)=e^{x}$ which is a $1-1$, strictly increasing function with domain, $\mathcal{D}=\mathbb{R}$ and range, $y>0$. Its inverse is given by the natural logarithm function denoted by $\ln$. The domain is all positive numbers and the range is all real numbers.

The natural logarithm is sometimes written as $\log _{e}$ or just log (often in programming languages). Generally, given $a>0$ and $x>0$ with $a \neq 1$, the logarithm base $a$ of $x$, written as $\log _{a} x$ gives the exponent to which $a$ needs to be raised to obtain $x$. That is, $\log _{a} x=y$ means $a^{y}=x$. It follows that $\log _{a} x$ and $a^{x}$ are inverses.

Back to the natural logarithm, its derivative is

Using Eq. (1.6), we write $y=e^{x}$ which is equivalent to $x=\ln y$,

Restricting domains

We mentioned that a function needs to be 1 - 1 to have an inverse. A non 1 - 1 function cannot have an inverse over its domain; however, we can define its inverse if we restrict the domain such that the function becomes 1 - 1 . For instance, $f(x)=x^{2}$ is not 1 - 1 for $x \in \mathbb{R}$. For the restricted functions $g(x)=x^{2}$ for $x \geq 0$ and $h(x)=x^{2}$ for $x \leq 0$, we can find an inverse.

Periodic functions such as the trigonometric functions sin, cos, tan cannot be 1 - 1 functions.

Definition 1.2 A function $f(x)$ is periodic with period $l$ if

for all $x$. The period of $f$ is the smallest positive $l$ for which the above relation holds.

To invert those functions, we restrict the domain. A conventional choice is to restrict the domain to $-\pi / 2 \leq x \leq \pi / 2$ for $\sin x$ and $0 \leq x \leq \pi$ for $\cos x$. Of course other domains can be used, e.g. $\pi / 2 \leq x \leq 3 \pi / 2$ giving different inverse trigonometric functions. The tangent function (see Fig. 1.3(a) for the graph of $y=\tan x$ in $[-2 \pi, 2 \pi]$ ) can also be inverted by restricting the domain, say to the open interval $-\pi / 2<x<\pi / 2$ [this is shown in red in Fig. 1.3(a)]. The values that form the restricted domain are called principal values. Figure 1.3(b) shows the inverse of $\tan x$ or $\arctan$ (given by $y=\tan ^{-1} x$ ) restricted in the open interval $(-\pi / 2, \pi / 2)$.

(a)

(b)

Figure 1.3: (a) The graph of $y=\tan x$ in $-2 \pi \leq x \leq 2 \pi$. The red part of the graph is in $-\pi / 2<x<\pi / 2$. (b) The inverse of the red part of graph in panel (a) given by $y=\tan ^{-1} x$; its range is $-\pi / 2<x<\pi / 2$.

Note on trigonometric functions

Recall the elementary geometric definition of $\operatorname{cosine}$ and sine which defines $\cos \theta$ and $\sin \theta$ for $0<\theta<\pi / 2$.

Using the right triangle shown, we define

$\sin \theta \quad$ We extend the definitions to the endpoints $\theta=0, \pi / 2$ as:

The sine and cosine satisfy the addition formulae,

which can be proven geometrically by stacking two right triangles.

Similarly, we can show,

It is much easier however to prove the addition formulae using complex numbers (see Chapter 7, Section 7.2) or rotation matrices. The addition formulae can be used to extend the definitions of $\cos \theta$ and $\sin \theta$ to any angle. The cosine and sine can also be defined (for any angle) through a polar form. Consider a circle of radius $r$ centred at the origin. The coordinates of the point $P$ on the circle are

where $\theta$ is the angle (taken anti-clockwise) between the $x$-axis and the line segment joining $\mathrm{O}$ and $\mathrm{P}$ (see below).

Using the polar form or the addition formulae.

Combining the two, we have

Similarly,

and consequently,

i.e. the tangent function is periodic with period $\pi$.