Q 114 - 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

114. $f\left(x\right)=\begin{array}{ll}\ln (-x)& \text{ if }x\le -1\\ -\ln (-x)& \text{ if }-1<x<0\end{array}$

Step-by-Step Solution

Answer

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

Step 1. Given information.

The following function is given $f\left(x\right)=\left\{\ln \left(-x\right):x\le \hspace{0.17em}-1,\hspace{0.17em}-\ln \left(-x\right):-1<\hspace{0.17em}x<\hspace{0.17em}0\right\}$.

Step 2. Domain and range.

Domain of $\left\{\ln \right(-x):x\le -1\}:x\le -1$

Domain of $\{-\ln (-x):-1<x<0\}: -1<x<0$

Combine domains of all the pieces:

$x\le -1$

or -1<x<0

x<0

Extreme Points of $\left\{\ln \right(-x):x\le -1,-\ln (-x):-1<x<0\}$ : Minimum (-1,0)

Find the range for the interval  $-\infty <x <-1 : 0< f\left(x\right)<\infty$

Find the range for the interval -1<x<0 : $0<f\left(x\right)<\infty$

Combine the ranges of all domain intervals to obtain the function range

$f\left(x\right)\ge 0$ or f(x)>0

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

Y- axis interception point of $\left\{\ln \right(-x):x\le -1,-\ln (-x):-1<x<0\}$ : None 

X-Intercepts: (-1,0)