Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter 3 - Section 3.2 - Exponential Functions - Exercise Set - Page 464: 110


Domain: $(-\infty,-2)\cup(6,\infty)$

Work Step by Step

Logarithmic functions are defined for positive arguments only. $ f(x)=\ln(x^{2}-4x-12)$ is defined for positive values. $ x^{2}-4x-12 \gt 0\quad $ Factor the trinomial -- find factors of $-12$ whose sum is $-4$; we find $-6$ and $+2.$ $(x+2)(x-6) \gt 0$ The graph of $ y=x^{2}-4x-12=(x+2)(x-6)$ is a parabola opening upwards, intersecting the x-axis at $-2$ and $+6.$ We know that $ y $ is positive where the graph is above the x-axis. The graph is above the x-axis before the left x-intercept $(x \lt -2) $ and to the right of the right intercept $(x \gt 6)$ Thus, the domain is: $(-\infty,-2)\cup(6,\infty)$
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