Lesson

Inverse functions have an application in everyday right-angled triangle trigonometry. If we know the ratio formed by two sides of a right-angled triangle, then we can use an inverse trigonometric function to find the angles.

In more general contexts, there are some things about inverse functions that must be treated with care.

The broad idea of an inverse function is that while a function maps each element $x$`x` in its domain to an element $y$`y` in its range, the inverse function reverses the process by mapping each $y$`y` back to the original $x$`x`.

This will only work if each $y$`y` in the range is the image of exactly one $x$`x` from the domain. Take for example the squaring function $y(x)=x^2$`y`(`x`)=`x`2 with its domain the real numbers. The number $9$9 is an element of the range but both $-3$−3 and $3$3 in the domain have this same number as their image. Therefore, we cannot know which was the original $x$`x`.

A more extreme example of a function that has no inverse is the constant function, $y(x)=k$`y`(`x`)=`k` for all real numbers $x$`x`. Whatever number $x$`x` we choose to start from, the function maps it to $k$`k`. If we are only given that the function value is $k$`k`, there is no way to tell which $x$`x` it came from. Hence, there can be no inverse function.

The so-called *horizontal line test* is used with graphical representations of a function to decide whether a function can have an inverse. If there exists a horizontal line that cuts the graph in more than one place, at least two elements of the domain have the same image under the function and there can be no inverse.

In the diagram, the horizontal line $y=0.25$`y`=0.25 cuts the graph in three places: $x_1$`x`1, $x_2$`x`2 and $x_3$`x`3. An inverse function would begin with the number $0.25$0.25 and map it to exactly one of the three possibilities, but there is no way to know which of them is the right one and, therefore, there can be no inverse function.

Functions that do have inverses are those that are something like a linear function, always increasing or always decreasing. In this situation, each element of the range is the image of exactly one domain element and so the inverse mapping is possible. Functions of this kind are called *one-to-one* functions.

We can often get around the problem of a function not being one-to-one and therefore having no inverse, by restricting the domain of the function. In the graph sketched above, we see that the function is decreasing in the central part of its domain between $-\frac{1}{\sqrt{3}}$−1√3 and $\frac{1}{\sqrt{3}}$1√3 so that if we only considered this part of the domain there would be no ambiguity about inverse mappings. The same function with the restricted domain is shown below.

In the case of the trigonometric functions, the horizontal line test indicates that none of the functions on its natural domain has an inverse. We deal with this problem by restricting the domains of the functions to regions over which the functions are one-to-one.

The sine function is increasing between $-\frac{\pi}{2}$−π2 and $\frac{\pi}{2}$π2. When the domain is restricted to this interval, the inverse function exists. It is called *arcsin*. The domain of the inverse sine function is the range of the sine function, the interval $[-1,1]$[−1,1], and the range of the inverse sine function is the restricted domain of sine function, the interval $[-\frac{\pi}{2},\frac{\pi}{2}]$[−π2,π2].

The cosine function is decreasing between $0$0 and $\pi$π. When the domain is restricted to this interval, the inverse function exists. It is called *arccos*. The domain of the inverse cosine function is the range of the cosine function, the interval $[-1,1]$[−1,1], and the range of the inverse cosine function is the restricted domain of cosine function, the interval $[0,\pi]$[0,π].

The tangent function is strictly increasing between $-\frac{\pi}{2}$−π2 and $\frac{\pi}{2}$π2. When the domain is restricted to this open interval, the inverse function exists. It is called *arctan*. The domain of the inverse tangent function is the range of the tangent function, the real numbers. The range of the inverse tangent function is the restricted domain of the tangent function, the interval $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$(−π2,π2).

A certain rectangle has sides $56$56 m and $33$33 m. What are the angles formed by the diagonal in the rectangle?

If we draw or imagine a diagram, we can see that one of the angles has tangent ratio $\frac{33}{56}$3356. Using a calculator, we evaluate $\tan^{-1}\left(\frac{33}{56}\right)$`t``a``n`−1(3356) and obtain the answer $30.51^\circ$30.51°. The other angle must be $90-30.51=59.49^\circ$90−30.51=59.49°.

The notation $\tan^{-1}$`t``a``n`−1 is used on calculators for the inverse tangent function, *arctan*.

Display and interpret the graphs of functions with the graphs of their inverse and/or reciprocal functions