To determine the horizontal asymptote of the function [tex]\( f(x) = 5^x - 1 \)[/tex], we need to analyze the behavior of the function as [tex]\( x \)[/tex] approaches positive and negative infinity.
1. Examine the behavior as [tex]\( x \to \infty \)[/tex]:
As [tex]\( x \)[/tex] increases towards positive infinity, the term [tex]\( 5^x \)[/tex] grows rapidly towards positive infinity because 5 is a constant greater than 1. Therefore, due to the subtraction of 1 from [tex]\( 5^x \)[/tex], [tex]\( f(x) \)[/tex] also approaches positive infinity. This does not affect our horizontal asymptote since we are looking for values that [tex]\( f(x) \)[/tex] approaches as [tex]\( x \)[/tex] becomes very large.
2. Examine the behavior as [tex]\( x \to -\infty \)[/tex]:
As [tex]\( x \)[/tex] decreases towards negative infinity, the term [tex]\( 5^x \)[/tex] approaches 0 because any positive base [tex]\( b \)[/tex] (where [tex]\( 0 < b < 1 \)[/tex]) raised to increasingly large negative powers tends to 0. So:
[tex]\[ 5^x \to 0 \][/tex]
When [tex]\( 5^x \to 0 \)[/tex], the function [tex]\( f(x) = 5^x - 1 \)[/tex] approaches:
[tex]\[ f(x) \to 0 - 1 = -1 \][/tex]
So, as [tex]\( x \to -\infty \)[/tex], [tex]\( f(x) \)[/tex] tends to [tex]\( -1 \)[/tex].
Thus, the horizontal asymptote for the graph of [tex]\( f(x) = 5^x - 1 \)[/tex] is:
[tex]\[ y = -1 \][/tex]
Therefore, the correct answer is:
[tex]\[ y = -1 \][/tex]
