To determine the end behavior of the function [tex]\( f(x) = \log_{10}(5x - 1) \)[/tex], we need to analyze what happens to the function as [tex]\( x \)[/tex] approaches positive and negative extremes.
1. Domain Analysis:
- The logarithmic function [tex]\( \log_{10}(y) \)[/tex] is defined only for [tex]\( y > 0 \)[/tex]. Therefore, the argument [tex]\( 5x - 1 \)[/tex] must be greater than 0.
- This gives the inequality [tex]\( 5x - 1 > 0 \)[/tex], which simplifies to [tex]\( x > \frac{1}{5} \)[/tex].
2. End Behavior as [tex]\( x \to \infty \)[/tex]:
- As [tex]\( x \to \infty \)[/tex]:
[tex]\( 5x - 1 \to \infty \)[/tex].
[tex]\(\log_{10}(5x - 1) \to \infty \)[/tex].
- So, [tex]\( f(x) \to \infty \)[/tex]. This means that as [tex]\( x \)[/tex] becomes very large, [tex]\( f(x) \)[/tex] increases without bound.
3. End Behavior as [tex]\( x \)[/tex] approaches the lower bound [tex]\( \frac{1}{5} \)[/tex] from the right:
- As [tex]\( x \to \frac{1}{5}^{+} \)[/tex]:
[tex]\( 5x - 1 \to 0^+ \)[/tex].
[tex]\(\log_{10}(5x - 1) \to -\infty \)[/tex].
- So, [tex]\( f(x) \to -\infty \)[/tex]. This means that as [tex]\( x \)[/tex] approaches [tex]\( \frac{1}{5} \)[/tex] from the right, [tex]\( f(x) \)[/tex] decreases without bound.
From the analysis, we see that as [tex]\( x \)[/tex] approaches positive infinity, the function increases without bound, and as [tex]\( x \)[/tex] approaches [tex]\( \frac{1}{5} \)[/tex] from the right, the function decreases without bound.
Therefore, the correct choice is:
A. One end increases and one end decreases
