Graphs of Simple Functions, their Inverses, and Compositions

As the last section concluded, we will be especially interested in solving equations -- especially polynomial equations -- going forward. One powerful strategy for solving equations involves using inverse functions. Indeed, this strategy has application to not only polynomials, but also equations involving a great many different functions.

As such, this section takes a deep look at several (simple) functions we might encounter in those aforementioned equations we might attempt to solve. Part of that "deep look" includes finding and examining the graphs for these functions -- and the graphs of their related inverse functions if they have them. As we work to understand more complicated functions, it will be especially important to understand how some "simple functions" affect the graphs of other functions when composed with them.

However, before getting into all that, let us clarify what type of inverses we are talking about here. Recall, we have seen a number of "inverses" at this point, so it will be good to be clear about which we currently intend.

All of the different types of inverses we have seen thus far have involved some operation that we sought to somehow "undo", returning things in some way to some appropriate "identity".

In the context of solving equations, this last (compositional) inverse of a function will be most useful.

As a quick example, the functions $f(x) = x^3$ and $g(x) = \sqrt[3]{x}$ are inverses (again, in a compositional sense) as $$\begin{array}{rcccccl} f(g(x)) &=& f(\sqrt[3]{x}) &=& (\sqrt[3]{x})^3 &=& x \quad \textrm{and}\\ g(f(x)) &=& g(x^3) &=& \sqrt[3]{x^3} &=& x \end{array}$$ As a matter of verbiage, we say a function $f$ is invertible if there exists a function $g$ such that $f$ and $g$ are inverses of one another.

As we have done previously, we denote the inverse of an invertible function $f$ by $f^{-1}$. That said, a common source of confusion for students stems from how similar this notation looks to the multiplicative inverse of the value $f(x)$ for some given $x$.

As the multiplicative inverse so-described potentially depends on the value of $x$, let us the adopt the following convention: When we write $f^{-1}(x)$, we will mean the (compositional) inverse of $f$, whereas when we write $[f(x)]^{-1}$, we will mean the multiplicative inverse of the value $f(x)$.

So as an example, suppose $f(x)=x^3$. Then, $$f^{-1}(x) = \sqrt[3]{x} \quad \quad \textrm{while,} \quad \quad [f(x)]^{-1} = \frac{1}{x^3}$$

There is more we can say about inverses, however -- especially with regard to their graphs.

Graphs of Invertible Functions and their Inverses

We have already noted that $f(x)=x^3$ is invertible with $f^{-1}(x) = \sqrt[3](x)$. Consider what happens when we draw the graphs of both of these functions on the same set of axes:

There is a striking symmetry between these two graphs -- one that is shared between any pair of inverse functions we might choose to draw. In every case, two functions that are inverses of one another will be symmetric about the identity function (whose graph is given by $y=x$).

To see why, note that $(x,y)$ is a point on the graph of some invertible function $f$ if and only if $(y,x)$ is a point on the graph of $y=f^{-1}(x)$.

As an example -- just consider the graphs of $f(x)=x^3$ (in red) and $f^{-1}(x)=\sqrt[3]{x}$ (in blue) shown above. The points $(-2,-8)$, $(-1,-1)$, $(0,0)$, $(1,1)$, and $(2,8)$ are all points on the graph of the former. While $(-8,-2)$, $(-1,-1)$, $(0,0)$, $(1,1)$, and $(8,2)$ are all on the graph of the latter.

To argue the more general case, suppose $(x_0,y_0)$ is a point on the graph of $y=f(x)$ and $f$ is invertible.

Consequently, $f(x_0) = y_0$, and thus, $f^{-1}(f(x_0)) = f^{-1}(y_0)$.

However, since $f^{-1}$ is the inverse of $f$, we also know $f^{-1}(f(x_0))=x_0$.

Hence, $f^{-1}(y_0) = x_0$, which tells us that $(y_0,x_0)$ is a point on the graph of $y=f^{-1}(x)$.

The Horizontal Line Test, Revisited

In our earlier discussion of functions, we noticed that a function must have two properties in order to have an inverse. It must be:

Recall that "injectivity" played out visually in the horizontal line test, where we noted that a function would not have an inverse when we could find some horizontal line that intersected the graph of the function two or more times, as this would represent one output value $y$ associated with two different inputs $x$.

We can make a parallel argument for why the horizontal line test works by appealing to this new-found symmetry of a function with its inverse across the line $y=x$.

Note that any function that fails the horizontal line test (like the red $f(x)=x^2$ below, given the green horizontal line) will have a reflection across the line $y=x$ (here, in blue), which must then intersect at least twice some vertical line (the reflection of the horizontal line, shown in orange for this example). The reflected graph of course can't correspond to any function as it has two $y$-values (outputs) associated with a single $x$-value (input)!

Some Simple Functions : their Graphs, Inverses, and Compositions

Let us not get ahead of our skis, however. Note that both applying the horizontal line test to determine if a function has an inverse and graphing the inverse (presuming it exists) by taking advantage of this symmetry over the line $y=x$ requires us to graph the original function in question.

Developing all of the skills needed to graph an arbitrary function takes time, however. Indeed, much of calculus is spent addressing this question. However, deducing the graphs of many simple functions is quite easy, and we do so next. Of course, with every graph of a function we determine, we get the graph of its inverse function (if it exists) immediately given the symmetry discussed above.

Consider the following types of functions -- in doing so, you may assume $c$ always represents some constant real value in all of the functions that follow: