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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 5: Q 45. (page 298)

Find the domain of the function.
$g (x) = \log_{5} (\frac{x + 1}{x})$

Short Answer

Expert verified

The domain of the given function $g (x)$ is $(- 鈭�, - 1) 鈭� (- 1, 0) 鈭� (0, 鈭�)$

Step by step solution

01

Step 1. Given information

A logarithmic function is given as $g (x) = \log_{5} (\frac{x + 1}{x})$

02

Step 2. Formula Used

The domain of the logarithmic function $f (x) = \log_{a} (x)$ is determined by $x > 0$ ,

03

Step 3. Finding The Domain

The function is $g (x) = \log_{5} (\frac{x + 1}{x})$

Now, for the domain,

role="math" localid="1648529671543" $\frac{x + 1}{x} > 0 \frac{x (x + 1)}{x^{2}} > 0$

On solving

$x (x + 1) > 0 x > 0 a n d x + 1 > 0 o r x < 0 a n d x + 1 < 0 x > 0 a n d x > - 1 o r x < 0 a n d x < - 1$

The required domain of $g (x)$ , using

interval notation, is $(- 鈭�, - 1) 鈭� (- 1, 0) 鈭� (0, 鈭�)$ .

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