Question

question 20 of 25
what can you say about the end behavior of the function
$f(x)=\log _{10}(5x - 1)$?
a. both ends decrease
b. both ends increase
c. one end decreases and one end approaches a constant
d. one end increases and one end decreases

Explanation:

Step1: Analyze the domain

The domain of \(y = \log_{10}(5x - 1)\) is \(5x-1>0\), i.e., \(x>\frac{1}{5}\).

Step2: Analyze the limit as \(x\to+\infty\)

As \(x\to+\infty\), \(5x - 1\to+\infty\). Then \(\lim_{x\to+\infty}\log_{10}(5x - 1)=+\infty\) (since \(\log_{a}t\) with \(a > 1\) is an increasing function and \(\lim_{t\to+\infty}\log_{a}t=+\infty\)).

Step3: Analyze the limit as \(x\to\frac{1}{5}^+\)

As \(x\to\frac{1}{5}^+\), \(5x - 1\to0^+\). Then \(\lim_{x\to\frac{1}{5}^+}\log_{10}(5x - 1)=-\infty\) (since \(\lim_{t\to0^+}\log_{a}t=-\infty\) for \(a>1\)).

Answer:

D. One end increases and one end decreases