Finding the inverse of a function and its graph.


Suppose we have a function f(x) and know that point (c, d) lies on its graph. If f(x) has an inverse, f -1(x), the point (d, c) lies on its graph. The points (c, d) and (d, c) are equidistant from the line y = x. Such points are called mirror images of each other. Notice the graph and the points with respect to the line y = x.







With respect to the line y = x, the points on the graphs of f(x) and f -1(x) are mirror images of each other, so it makes sense to draw the conclusion that the graphs of f(x) and f -1(x) are also mirror images of each other with respect to y = x
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Examples: a. Find the inverse of f(x) = 2x + 4 and graph f(x) and f -1(x) on one rectangular coordinate system.

Before we find the inverse of f(x), we must determine if an inverse exists. If the function is one-to-one, then it has an inverse. Since f(x) is linear, then it does have an inverse. Recall, to find the inverse we interchange y and x and then solve the equation for y.

f(x) = 2x + 4
y = 2x + 4
x = 2y + 4
x – 4 = 2y
(x – 4)/2 = y
f -1(x) = (x – 4)/2

To graph f(x) = 2x + 4, we know the y- intercept is (0, 4). To find the x- intercept, rewrite the function as the equation y = 2x + 4 and substitute 0 for y. Then solve for x.

y = 2x + 4
0 = 2x + 4
-4 = 2x
-2 = x. Therefore the x- intercept is (-2, 0).

To graph f -1(x) = (1/2)x – 2, we know the y- intercept is (0, -2). To find the x- intercept, rewrite the inverse function as the equation y = (1/2)x – 2 and substitute 0 for y. Then solve for x.

y = (1/2)x – 2
0 = (1/2)x – 2
2 = (1/2)x, 4 = x. Therefore the x- intercept is (4, 0)

Notice the graph of f(x) and its inverse.