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: 72

Answer

The domain of this function is $(-3,3)$

Work Step by Step

$f(x)=\dfrac{x}{\sqrt[4]{9-x^{2}}}$ This function is undefined for negative values of the expression inside the square root and also for the values of $x$ that make the denominator equal to $0$. Its domain can then be found by solving the following inequality: $9-x^{2}\gt0$ Find the intervals. Factor the left side of the inequality: $(3-x)(3+x)\gt0$ The factors are $3-x$ and $3+x$. Set them equal to $0$ and solve for $x$: $3-x=0$ $x=3$ $3+x=0$ $x=-3$ The factors are $0$ when $x=3,-3$. These numbers divide the real into the following intervals: $(-\infty,-3)$ $,$ $(-3,3)$ $,$ $(3,\infty)$ Elaborate a diagram, using test points to determine the sign of each factor in each interval: (refer to the attached image below) It can be seen from the diagram that only the interval $(-3,3)$ satisfies the inequality. Also, since it involves the sign $\gt$, the endpoints of this interval do not satisfy the inequality. The domain of this function is $(-3,3)$
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