Question
Identify the horizontal asymptote of f(x) = 3 over 5 x.
y = 3 over 5
y = 0
y = 5 over 3
No horizontal asymptote
y = 3 over 5
y = 0
y = 5 over 3
No horizontal asymptote
Answers
Answered by
GPT-4o mini
To find the horizontal asymptote of the function \( f(x) = \frac{3}{5} x \), we need to analyze the behavior of the function as \( x \) approaches infinity (both positive and negative).
1. The function \( f(x) = \frac{3}{5} x \) is a linear function, and as \( x \) becomes very large (either positively or negatively), the function value will also become very large (positively or negatively).
2. Horizontal asymptotes are found by determining if \( f(x) \) approaches a specific constant value as \( x \) approaches \( \infty \) or \( -\infty \).
In this case, as \( x \to \infty \) or \( x \to -\infty \):
\[
f(x) = \frac{3}{5} x \to \infty \quad \text{(or)} \quad f(x) \to -\infty
\]
Since \( f(x) \) does not approach a constant value, we conclude that there is no horizontal asymptote.
Thus, the answer is:
**No horizontal asymptote.**
1. The function \( f(x) = \frac{3}{5} x \) is a linear function, and as \( x \) becomes very large (either positively or negatively), the function value will also become very large (positively or negatively).
2. Horizontal asymptotes are found by determining if \( f(x) \) approaches a specific constant value as \( x \) approaches \( \infty \) or \( -\infty \).
In this case, as \( x \to \infty \) or \( x \to -\infty \):
\[
f(x) = \frac{3}{5} x \to \infty \quad \text{(or)} \quad f(x) \to -\infty
\]
Since \( f(x) \) does not approach a constant value, we conclude that there is no horizontal asymptote.
Thus, the answer is:
**No horizontal asymptote.**