Calculus with Applications (10th Edition)

Published by Pearson
ISBN 10: 0321749006
ISBN 13: 978-0-32174-900-0

Chapter 2 - Nonlinear Functions - 2.1 Properties of Functions - 2.1 Exercises - Page 54: 24


$\left[ \frac{-5}{3},\infty\right)$

Work Step by Step

We want to determine all real $x$ values for which $f(x)=(3x+5)^{1/2}$ is defined. We can rewrite $f(x)$ as $$f(x)=\sqrt{3x+5}.$$ We can only take the square root of a nonnegative real number, so set the inside of the square root greater than or equal to 0 to find all $x$ where $f$ is defined: $$3x+5 \geq 0.$$ Subtracting 5 from both sides of the inequality gives: $$3x \geq -5$$ Divide both sides by 3 to obtain: $$x \geq \frac{-5}{3}$$. Therefore, the domain of $f$ is $x \geq \frac{-5}{3}$, which can be written as $\left[ \frac{-5}{3},\infty\right)$ in interval notation.
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