To determine the horizontal asymptote and the [tex]\( y \)[/tex]-intercept of the function [tex]\( f(x) = 6^x \)[/tex], follow these steps:
### Horizontal Asymptote
1. Definition of Asymptote: A horizontal asymptote is a horizontal line that the graph of a function approaches as the input ([tex]\( x \)[/tex]) either goes to positive infinity ([tex]\( x \to \infty \)[/tex]) or negative infinity ([tex]\( x \to -\infty \)[/tex]).
2. Behavior of [tex]\( 6^x \)[/tex] as [tex]\( x \to \infty \)[/tex]: Notice that as [tex]\( x \to \infty \)[/tex], [tex]\( 6^x \)[/tex] grows without bound, becoming very large.
3. Behavior of [tex]\( 6^x \)[/tex] as [tex]\( x \to -\infty \)[/tex]: As [tex]\( x \to -\infty \)[/tex], [tex]\( 6^x \)[/tex] approaches 0 because any positive base raised to an increasingly negative power will approach zero.
Thus, the horizontal asymptote is [tex]\( y = 0 \)[/tex].
### [tex]\( y \)[/tex]-Intercept
1. Definition of [tex]\( y \)[/tex]-Intercept: The [tex]\( y \)[/tex]-intercept is the point where the graph of the function crosses the [tex]\( y \)[/tex]-axis. This occurs when [tex]\( x = 0 \)[/tex].
2. Calculation of [tex]\( y \)[/tex]-Intercept: Evaluate [tex]\( f(0) \)[/tex]:
[tex]\[
f(0) = 6^0 = 1
\][/tex]
Therefore, the function crosses the [tex]\( y \)[/tex]-axis at the point [tex]\( (0, 1) \)[/tex].
The correct answers are:
- Horizontal asymptote: [tex]\( y = 0 \)[/tex]
- [tex]\( y \)[/tex]-intercept: [tex]\( (0, 1) \)[/tex]
Given the options, the correct choice is:
B. asymptote: [tex]\( y = 0 \)[/tex]
[tex]\( y \)[/tex]-intercept: [tex]\( (0, 1) \)[/tex]
