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Example 2 of finding the function f(x) Using the variable substitution method

In this example, we are given the function f(1- sin x) = sin2x, the question ask for f(x). How to find the f(x)? We are using the variable substitution method. Let the variable t = 1 - sin x, then sin x = 1 - t, sin2x = (1 - t)2. What is the range of t? Since the maximum value of sin x is 1, when sin x = 1, t = 1 - sin x = 1 - 1 = 0. Since the minimum value of the sin x is -1, when sin x = -1, t = 1 - six x = 1 - (-1) = 1 + 1 = 2. So the range of t is from 0 to 2. Now we substitute 1 - sin x to t, we get the function f(t) = (1 - t)2 and its domain is t in the range of [0, 2]. Now change the variable t to x, we get f(x) = (1 - x)2 and its domain is x in the range of [0. 2].