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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 4: Q. 69 (page 242)

What is the domain of the function $f (x) = \sqrt{x^{4} - 16}$ ?

Short Answer

Expert verified

The domain of the function is ${x | x 鈮� - 4 o r x 鈮� 4}$ and the interval notation is $(- 鈭�, - 4] 鈭� [4, 鈭�) .$

Step by step solution

01

Step 1. Given Information

The given function is $f (x) = \sqrt{x^{4} - 16}$

We have to find the domain.

02

Step 2. Finding the domain

As the given function is a square root function.

To find the domain of a square root function, set the radicand to greater than or equal to zero.

So,

$x^{4} - 16 鈮� 0 (x^{2} - 4) (x^{2} + 4) 鈮� 0$

03

Step 3. Check the points

Let $x = - 6$ and substitute in $(x^{2} + 4) (x^{2} - 4) 鈮� 0$

$({(- 6)}^{2} + 4) ({(- 6)}^{2} - 4) 鈮� 0 (40) (32) 鈮� 0$

So, it is true.

Let $x = 0$ and substitute in $(x^{2} + 4) (x^{2} - 4) 鈮� 0$

$(0 + 4) (0 - 4) 鈮� 0 - 16 鈮� 0$

So, it is false.

Let $x = 6$ and substitute in $(x^{2} + 4) (x^{2} - 4) 鈮� 0$

$({(6)}^{2} + 4) ({(6)}^{2} - 4) 鈮� 0 (40) (32) 鈮� 0$

So, it is true.

The domain of the function is ${x | x 鈮� - 4 o r x 鈮� 4}$ and the interval notation is $(- 鈭�, - 4] 鈭� [4, 鈭�) .$

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