Q 113 - Precalculus Enhanced with Graphing Utilities

Question

In Problems 113-116, graph each function. Based on the graph, state the domain and the range and find any intercepts

113. $$\begin{gathered} f\left(x\right)=\ln (-x) if x<0 \\ =\ln \left(x\right) if x>0 \end{gathered}$$

Step-by-Step Solution

Answer

The domain is $(-\infty ,0)\cup (0,\infty )$ and the range is $(-\infty ,\infty )$ and the X- axis intercept is $(-1,0),(1,0)$.

Step 1. Given information.

The following function is given $$\begin{gathered} f\left(x\right)=\ln (-x) if x<0 \\ =\ln \left(x\right) if x>0 \end{gathered}$$.

Step 2. The range and domain.

Domain of $\left\{\ln \right(-x):x<0\}: x<0$

Domain of $\left\{\ln \right(x):x>0\}: x>0$

Hence, the domain is $(-\infty ,0)\cup (0,\infty )$.

Extreme Points of $\left\{\ln \right(-x):x<0,\ln (x):x>0\}$ : None

Find the range for the interval $-\infty <x<0: -\infty <f\left(x\right)<\infty$

Find the range for the interval $0<x<\infty : -\infty <f\left(x\right)<\infty$

Combine the ranges of all domain intervals to obtain the function range $-\infty <f\left(x\right)<\infty$

X-axis interception points of $\left\{\ln \right(-x):x<0,\ln (x):x>0\}: (-1,0),(1,0)$

Y-axis interception point of $\left\{\ln \right(-x):x<0,\ln (x):x>0\}$: None

X Intercepts: (-1,0),(1,0)