By Marcel Finan
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Additional resources for A semester course in trigonometry
When a function has an inverse then we say that the function is invertible. 1 Find the inverse function of (a) f (x) = log x (b) g(x) = ex . Solution. (a) f −1 (x) = 10x (b) g −1 (x) = ln x. 1 It is important not to confuse between f −1 (x) and (f (x))−1 . The later is just 1 whereas the former is how the the reciprocal of f (x), that is, (f (x))−1 = f (x) inverse function is represented. 1 shows the relationship between f and f −1 . 1 This figure shows that we get the inverse of a function by simply reversing the direction of the arrows.
We see that the new graph is obtained from the old one by shifting to the left 1 unit. This is as expected since the value of x2 is the same as the value of (x + 1)2 at the point 1 unit to the left. 8 that we get the graph of h(x) by moving the graph of f (x) to the right 1 unit. 8 In general, if c > 0, the graph of f (x + c) is obtained by shifting the graph of f (x) to the left a distance of c units. The graph of f (x − c) is obtained by shifting the graph of f (x) to the right a distance of c units.
1 Graph the functions f (x) = 2x and −f (x) = −2x on the same axes. Solution. 1. 1 Reflection About the y-Axis We know that the points (x, f (x)) and (−x, f (x)) are on opposite sides of the y-axis. So the graph of the new function f (−x) is the reflection of the graph of f (x) about the y-axis. 2 Graph the functions f (x) = x3 and f (−x) = −x3 on the same axes. Solution. 2. 2 Symmetry About the y-Axis When constructing the graph of f (−x) sometimes you will find that this new graph is the same as the graph of the original function.