1.5.4 Graphical Representation of Inverse Functions

The graphs of a function and its inverse have a special relationship: For each point `text[(]a,btext[)]` on the graph of `f`, there is a point `text[(]b,atext[)]` on the graph of the inverse of `f`. The points `text[(]a,btext[)]` and `text[(]b,atext[)]` are mirror images in the line `y=x` (see Figure 4). Thus the graph of `f` can be reflected about the line `y=x` to obtain the graph of the inverse of `f`. Notice that this reflection takes the `y`-axis to the `x`-axis and vice versa; that is, it interchanges `x`- and `y`-coordinates.

We illustrate this mirroring with an entire figure (not just a point) by repeating Figure 3 along with its reflection in Figures 5 and 6. Here we have taken "0" on the time scale to mean 1980. Since the variables have different units, we have made 20 years equivalent to 5 QBtu, an arbitrary choice that is convenient for this example.

Figure 5   Iran's Energy Consumption, 1981-2000	Figure 6   Time as a function of consumption

Checkpoint 1

Following the same geometric idea for inverting functions, we show in Figures 7 and 8 the graphs of `y^2=x` and `y=x^2`. Each of these equations is obtained from the other by interchanging `x` and `y`. Each of the graphs is obtained from the other by reflecting in the line `y=x`, which amounts to the same thing.

Figure 7   Positive and negative square roots	Figure 8   Squaring positive and negative numbers (mirroring Figure 7 in the line y = x)

However, these figures reveal a new complication, suggested by the blue and green parts of the two graphs. The upper (blue) part of the graph of `y^2=x` is also the graph of the (positive) square root function, `y=sqrtx`. The inverse of this function is not the entire squaring function but only the part shown with a blue curve in Figure 8. That is, the inverse of the square root function is the function defined by `y=x^2`, `x>=0`. Since the square root function has only positive numbers (and `0`) for its outputs, its inverse has only positive numbers (and `0`) for its inputs.

Checkpoint 2

Suppose we focus now on the squaring function in Figure 8. The domain for this function consists of all real numbers `x` — positive, negative, and zero. That is, the entire graph in Figure 8, blue and green, is the graph of the squaring function. What's the inverse of this function? Well, if we invert all the pairs of numbers that represent points on the graph, we get all the points of the graph in Figure 7 — blue and green. That's a perfectly good relation, but it's not a function. (Why? Refer back to your criterion in Activity 1 of Section 1.4.) That's the new complication: The inverse of a function may not be a function!

However, we can usually produce a function whose inverse is also a function if we restrict the domain in some appropriate way — such as considering the squaring function only for positive numbers. Your computer algebra system already knows this, as we will see when we study its various inverse functions.