Q 45. - Precalculus Enhanced with Graphing Utilities

Question

Find the domain of the function.

$g\left(x\right)={\log }_5\left(\frac{x+1}{x}\right)$

Step-by-Step Solution

Answer

The domain of the given function $g\left(x\right)$ is $\left(-\infty ,-1\right)\cup \left(-1,0\right)\cup \left(0,\infty \right)$

Step 1. Given information

A logarithmic function is given as $g\left(x\right)={\log }_5\left(\frac{x+1}{x}\right)$

Step 2. Formula Used

The domain of the logarithmic function $f\left(x\right)={\log }_a\left(x\right)$ is determined by $x>0$,

Step 3. Finding The Domain

The function is $g\left(x\right)={\log }_5\left(\frac{x+1}{x}\right)$

Now, for the domain,

$$\begin{gathered} \frac{x+1}{x}>0 \\ \frac{x\left(x+1\right)}{x^2}>0 \end{gathered}$$

On solving

$$\begin{gathered} x\left(x+1\right)>0 \\ x>0 and x+1>0 or x<0 and x+1<0 \\ x>0 and x>-1 or x<0 and x<-1 \end{gathered}$$

The required domain of $g\left(x\right)$, using

interval notation, is $\left(-\infty ,-1\right)\cup \left(-1,0\right)\cup \left(0,\infty \right)$.