6.2 Inverse Functions
1.
Consider the graph of a strictly monotonic function and its inverse (Fig. 1.a). Notice that f −1 (x) is
the reflection of f (x) about the y = x line
(a) Some points along the f (x) graph are given. Exchange the x and y coordinates for each (x, y) pair.
Plot your results on the grid (Fig. 1.a). Notice that (9,5) becomes (5,9), for example. Convince
yourself that each point on the f (x) graph has a mirror reflection point on the f −1 (x) graph.
(b) In a test situation, you might be asked for a crude sketch. Starting from a crude sketch of the
function (Fig. 1.b), make a crude sketch of the inverse, on the same graph.
(c) For the function, f (x), notice that points (−1, −8), (0, −6) and (2, −2) fall along a straight line.
Use this observation to compute f ′ (0) or, in other words, compute the derivative of the function
f (x) at at the point (0, −6). The slope m is given by m = (y1 − y0 )/(x1 − x0 ), for a straight line.
(d) Now, using your sketch of the inverse, from part (b), compute (f −1 )′ (−6) or, in other words,
compute the derivative of the inverse f −1 (x) at the point (−6, 0). Use the slope again.
(e) From parts (c) and (d), compare the derivative of the inverse at (−6, 0) to the derivative of the
function at (0, −6) and notice that (f −1 )′ (−6) equals the reciprocal of f ′ (0).
(f ) Now suppose that you do not have a sketch of the inverse (as in problem 37 from page 336 of your
book). To find (f −1 )′ (−6) you can look for the value of x such that y = f (x) = −6, which gives
x = 0 in this example. Then you can put x = 0 and y = −6 into the following formula:
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(f −1 )′ (y) = ′
f (x)
which is given on page 335 of your book. Compare to your reslut from part (d).
10
10
f(x)
8
Crude Sketch
8
y=x
y=x
f−1(x)
6
6
(9,5)
4
(9,5)
4
(8,2)
(8,2)
2
2
(5,0)
y
y
(5,0)
0
−2
0
−2
(2,−2)
(2,−2)
−4
−4
−6
−6
(0,−6)
(0,−6)
−8
−8
(−1,−8)
−10
−10
−8
−6
−4
−2
0
x
2
(−1,−8)
4
6
8
−10
−10
10
(a) function (solid line) and inverse (dashed line)
−8
−6
−4
−2
0
x
2
4
6
8
10
(b) crude sketch of graph
Figure 1: This figure shows the graphs of a function (a, solid line) and a crude sketch (b) of a strictly
monotonic function. The graph of the inverse is shown as well (a, dashed line).
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