Example 5
Sketch the graph of the inverse of the function f whose graph is shown in Fig. 1.49.
Solution 5
By inspection of the graph we see that f is onetoone (i.e., takes different domain values to different range values) and onto (i.e., takes on all values in the range) from S = [−2, 3] to T = [1, 5]. Therefore f has an inverse. The graph of f ^{−1} is exhibited in Fig. 1.50.
You Try It: Sketch f ( x ) = x^{3} + x and its inverse.
Another useful fact is this: Since an invertible function must be onetoone, two different x values cannot correspond to (that is, be “sent by the function to”) the same y value. Looking at Figs. 1.51 and 1.52, we see that this means
In order for f to be invertible, no horizontal line can intersect the graph of f more than once.
In Fig. 1.51, the fact that the line y = 2 intersects the graph twice means that the function f takes the value 2 at two different points of its domain (namely at x = −2 and x = 6). Thus f is not onetoone so it cannot be invertible. Figure 1.52 shows what happens if we try to invert f : the resulting curve is not the graph of a function.
Example 6
Look at Figs. 1.53 and 1.55. Are the functions whose graphs are shown in parts (a) and (b) of each figure invertible?
Solution 6
Graphs (a) and (b) in Fig. 1.53 are the graphs of invertible functions since no horizontal line intersects each graph more than once. Of course we must choose the domain and range appropriately. For (a) we take S = [−4, 4] and T = [−2, 3]; for (b) we take S = (−3, 4) and T = (0, 5). Graphs (a) and (b) in Fig. 1.54 are the graphs of the inverse functions corresponding to (a) and (b) of Fig. 1.53 respectively. They are obtained by reflection in the line y = x.
In Fig. 1.55 , graphs (a) and (b) are not the graphs of invertible functions. For each there is exhibited a horizontal line which intersects the graph twice. However graphs (a) and (b) in Fig. 1.56 exhibit a way to restrict the domains of the functions in (a) and (b) of Fig. 1.55 to make them invertible. Graphs (a) and (b) in Fig. 1.57 show their respective inverses.
You Try It: Give an example of a function from to that is not invertible, even when it is restricted to any interval of length 2.
Practice problems for this concept can be found at: Calculus Basics Practice Test.
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