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By Marcel Finan

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In this section we want to study the effect of multiplying a function by a constant k. This will result by either a vertical stretch or vertical compression. A vertical stretching is the stretching of the graph away from the x-axis. A vertical compression is the squeezing of the graph towards the x-axis. 10 (a) Complete the following tables x y = x2 -3 -2 -1 0 1 2 3 x y = 2x2 -3 -2 -1 0 1 2 3 x y = 3x2 -3 -2 -1 0 1 2 3 (b) Use the tables of values to graph and label each of the 3 functions on the same axes.

3 (e) f (x) = |5−x| , g(x) = − x2 . 5 Evaluate each composite function where f (x) = 2x + 3, g(x) = x2 − 5x, and h(x) = 4 − 3x2 . (a) (f ◦ g)(−3) (b) (h ◦ g) 2 5 √ (c) (g ◦ f )( 3) (d) (g ◦ f )(2c). 64 6 Inverse Functions An important feature of one-to-one functions is that they can be used to build new functions. So suppose that f is a one-to-one function. A new function, called the inverse function (denoted by f −1 ), is defined such that if f takes an input x to an output y then f −1 takes y as its input and x as its output.

Solution. (a) (f ◦ g)(x) = f (g(x)) = f (x2 − 3) = 2(x2 − 3) + 1 = 2x2 − 5. Similarly, (g ◦ f )(x) = g(f (x)) = g(2x + 1) = (2x + 1)2 − 3 = 4x2 + 4x − 2. (b) (f ◦ g)(5) = 2(5)2 − 5 = 45 and (g ◦ f )(−3) = 4(−3)2 + 4(−3) − 2 = 22. (c) f ◦ g = g ◦ f. With only one function you can build new functions using composition of the function with itself. Also, there is no limit on the number of functions that can be composed. 5 Suppose that f (x) = 2x + 1 and g(x) = x2 − 3. (a) Find (f ◦ f )(x). (b) Find [f ◦ (f ◦ g)](x).

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A semester course in trigonometry by Marcel Finan


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