Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning

Chapter 1 - Section 1.5 - Inverse Functions and Logarithms - 1.5 Exercises: 2

Answer

(a) The inverse function is defined as follows: $$y=f(x)\Rightarrow x=f^{-1}(y).$$ Its domain is $B$ and its range is $A$. (b) If you have the formula for $f(x)$, then you solve the equation $$y=f(x)$$ for $x$: $$x=f^{-1}(y)$$ (c) You symmetrically reflect the graph with respect to the line $y=x$.

Work Step by Step

(a) The inverse function is defined as follows: If the value of function $f$ at the point $x$ is $y$ then the value of the function $f^{-1}$ at the point $y$ is $x$ i.e. it satisfies: $$y=f(x)\Rightarrow x=f^{-1}(y).$$ Its domain is the range of the function $f$ which is $B$ and its range is the domain of the function $f$ which is $A$. (b) If you have the formula for $f(x)$ then you solve the equation $$y=f(x)$$ for $x$ and the expression you get for it: $$x=f^{-1}(y)$$ is the inverse function. (c) You symmetrically reflect the graph with respect to the line $y=x$ i.e. the straight line that passes through the origin at $45^\circ$ with respect to $x$ axis (and $y$ axis).

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