Calculus with Applications (10th Edition)

Published by Pearson

Chapter 2 - Nonlinear Functions - 2.1 Properties of Functions - 2.1 Exercises: 25

Answer

$(-\infty, -1) \cup (-1,1) \cup (1, \infty)$

Work Step by Step

We want to determine all real $x$ values for which $f(x)=\frac{2}{1-x^2}$ is defined. We can never divide by 0, so set the denominator of $f$ equal to 0 to find all $x$ values that would force us to divide by 0 if plugged into $f$: $$1-x^2=0.$$ Since the left side of the equation is a difference of squares, we can factor to obtain: $$(1-x)(1+x)=0$$. Setting each factor equal to 0 and solving for $x$ gives $x=1$ and $x=-1$. Since these $x$ values give us 0 in the denominator, these are the only two values for which $f$ is not defined. Therefore, the domain of $f$ is all real numbers except $x=1$ and $x=-1$. This can be written in interval notation as $(-\infty, -1) \cup (-1,1) \cup (1, \infty).$

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