College Algebra (10th Edition)

Published by Pearson
ISBN 10: 0321979478
ISBN 13: 978-0-32197-947-6

Chapter 6 - Section 6.2 - One-to-One Functions; Inverse Functions - 6.2 Assess Your Understanding - Page 421: 53

Answer

(a) $f^{-1}(x)=\dfrac{x-2}{4}$ (b) The domain and range of $f(x)$ and $f^{-1}(x)$ are both the set of real numbers. (c) Refer to the graph below ($f(x)$ is blue, $f^{-1}(x)$ is green, $y=x$ is black).
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Work Step by Step

(a) To find the inverse of the given function, perform the following steps: 1. Replace $f(x)$ with $y$. $$y=4x+2$$ 2. Interchange $x$ and $y$. $$x=4y+2$$ 3. Solve for $y$. \begin{align*} x-2&=4y\\ \frac{x-2}{4}&=\frac{4y}{4}\\ \frac{x-2}{4}&=y\\ y&=\frac{x-2}{4} \end{align*} 4. Replace $y$ with $f^{-1}(x)$. $$f^{-1}(x)=\dfrac{x-2}{4}$$ (b) The given function has the set of real numbers as its domain and range. Thus, its inverse will also have the set of real numbers as its domain and range. (c) Use a graphing utility to graph $f(x)$ (blue graph) $f^{-1}(x)$ (green graph) and $y=x$ (black graph). Refer to the graphs below.
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