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. Behavior as [tex]\( x \)[/tex] approaches positive infinity:
- For [tex]\( f(x) = 5^x - 1 \)[/tex], as [tex]\( x \)[/tex] goes to [tex]\( +\infty \)[/tex], the term [tex]\( 5^x \)[/tex] grows exponentially, meaning it increases without bound.
- Therefore, [tex]\( 5^x \)[/tex] becomes very large, and when we subtract 1, it still remains very large. So, [tex]\( f(x) \rightarrow \infty \)[/tex].
2. Behavior as [tex]\( x \)[/tex] approaches negative infinity:
- For [tex]\( f(x) = 5^x - 1 \)[/tex], as [tex]\( x \)[/tex] goes to [tex]\( -\infty \)[/tex], the term [tex]\( 5^x \)[/tex] becomes very small because raising 5 to a large negative power results in a fraction close to zero.
- Therefore, [tex]\( 5^x \)[/tex] approaches 0. So, the function [tex]\( f(x) = 5^x - 1 \)[/tex] approaches [tex]\( 0 - 1 = -1 \)[/tex].
So, as [tex]\( x \)[/tex] goes to [tex]\( -\infty \)[/tex], [tex]\( f(x) \)[/tex] approaches [tex]\(-1\)[/tex].
Therefore, the horizontal asymptote of the function [tex]\( f(x) = 5^x - 1 \)[/tex] is [tex]\( y = -1 \)[/tex].
The answer is:
[tex]\[ y = -1 \][/tex]
