Precalculus: Mathematics for Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 1305071751
ISBN 13: 978-1-30507-175-9

Chapter 2 - Section 2.1 - Functions - Exercises - Page 157: 56

Answer

$\{x\in R: x \neq 2\}=(-\infty , 2) \cup (2, \infty)$

Work Step by Step

The domain of $f(x)=\frac{1}{3x-6}$ is the set of all real numbers $x$ for which $\frac{1}{3x-6}$ is defined as a real number. We know a rational expression is not defined where the denominator is zero, so we must find all values of $x$ for which $$3x-6 =0.$$ We add $6$ to both sides to get $$3x=6,$$ and dividing both sides by $3$ gives $$x=2.$$ Thus we know $\frac{1}{3x-6}$ is not defined when $x=2$. This means the domain of $f(x)=\frac{1}{3x-6}$ is the set af all real numbers $x$ for which $x\neq 2$ In set notation, the domain is $\{x\in R: x \neq 2\}.$ In interval notation, the domain is $(-\infty , 2) \cup (2, \infty).$
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