Intermediate Algebra: Connecting Concepts through Application

Published by Brooks Cole
ISBN 10: 0-53449-636-9
ISBN 13: 978-0-53449-636-4

Chapter 7 - Rational Functions - 7.1 Rational Functions and Variation - 7.1 Exercises - Page 566: 43


$\text{all real numbers except }$ $ t=\left\{ -7,7 \right\} $

Work Step by Step

$\bf{\text{Solution Outline:}}$ The domain of the given rational function, $ h(t)=\dfrac{t+7}{t^2-49} ,$ are the values of $ t $ which will NOT make the denominator equal to $0.$ $\bf{\text{Solution Details:}}$ The values of $ t $ that will make the denominator equal to $0$ are \begin{array}{l}\require{cancel} t^2-49=0 .\end{array} Using the factoring of the difference of $2$ squares, which is given by $a^2-b^2=(a+b)(a-b),$ the equation above is equivalent to \begin{array}{l}\require{cancel} (t+7)(t-7)=0 .\end{array} Equating each factor to zero (Zero Product Property), then \begin{array}{l}\require{cancel} t+7=0 \\\\\text{OR}\\\\ t-7=0 .\end{array} Solving each equation results to \begin{array}{l}\require{cancel} t+7=0 \\\\ t=-7 \\\\\text{OR}\\\\ t-7=0 \\\\ t=7 .\end{array} Hence, the domain is the set of $ \text{all real numbers except }$ $ t=\left\{ -7,7 \right\} .$
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