AP® Precalculus · Free response

A trigonometric function and a graph: composites, every solution, and an inverse

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Q1: Function Concepts · Part A: graphing calculator allowed · 6 points

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The question

Name: ______________________   Date: ____________   A trigonometric function and a graph: composites, every solution, and an inverse (6 points)

Graph

The graph of ff, and g(x)=2sin⁡(πx4)+1g(x) = 2\sin\left(\frac{\pi x}{4}\right) + 1

The graph of the function ff on [0,8][0, 8] is made of three line segments, through the points (0,1)(0, 1), (2,5)(2, 5), (5,−1)(5, -1) and (8,2)(8, 2). The function gg is given by g(x)=2sin⁡(πx4)+1g(x) = 2\sin\left(\frac{\pi x}{4}\right) + 1. A graphing calculator is allowed; use radian mode.

12345678−2−1123456xyf

  1. (A(i)) Find the value of f(g(2))f(g(2)). Show the work that leads to your answer. [1 point]

  2. (A(ii)) Find every value of xx in [0,8][0, 8] for which f(x)=1f(x) = 1. [1 point]

  3. (B) Find every value of xx in [0,8][0, 8] for which g(x)=2.5g(x) = 2.5. Give each answer correct to three decimal places, and show the setup. [2 points]

  4. (C(i)) Is gg invertible on the interval [0,4][0, 4]? Give a reason for your answer using values of gg. [1 point]

  5. (C(ii)) On the interval [−2,2][-2, 2], gg is invertible. Write an expression for g−1(x)g^{-1}(x), for −1≤x≤3-1 \le x \le 3. [1 point]