University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 1 - Section 1.6 - Inverse Functions and Logarithms - Exercises - Page 49: 38


- Figure out the formula of the inverse of $f(x)=mx+b$ with $m\ne0$. - Looking at the formula, we can deduce the shape of the graph, as well as its slope and its $y$-intercept.

Work Step by Step

$$y=f(x)=mx+b\hspace{1cm}m\ne0$$ (a) To find its inverse: 1) Solve for $x$ in terms of $y$: $$y=mx+b$$ $$mx=y-b$$ $$x=\frac{y-b}{m}$$ 2) Interchange $x$ and $y$: $$y=\frac{x-b}{m}$$ Therefore, $$f^{-1}(x)=\frac{x-b}{m}$$ Here, we can rewrite the formula of the inverse: $$f^{-1}(x)=\frac{1}{m}x-\frac{b}{m}$$ Looking at the formula, we can conclude that the graph of the inverse is a line, with the slope $1/m$. The $y$-intercept is $-b/m$, as expected by the exercise.
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