Let's determine the [tex]$y$[/tex]-intercept and the horizontal asymptote for the function [tex]\( g(x) = 3^x + 4 \)[/tex].
### Finding the [tex]$y$[/tex]-intercept
The [tex]$y$[/tex]-intercept occurs where [tex]\( x = 0 \)[/tex]. To find the [tex]$y$[/tex]-intercept, evaluate [tex]\( g(x) \)[/tex] at [tex]\( x = 0 \)[/tex]:
[tex]\[ g(0) = 3^0 + 4 \][/tex]
Since [tex]\( 3^0 = 1 \)[/tex]:
[tex]\[ g(0) = 1 + 4 = 5 \][/tex]
Thus, the [tex]$y$[/tex]-intercept of the function is [tex]\( (0, 5) \)[/tex].
### Finding the horizontal asymptote
To determine the horizontal asymptote, we need to analyze the behavior of [tex]\( g(x) = 3^x + 4 \)[/tex] as [tex]\( x \)[/tex] approaches positive and negative infinity.
1. As [tex]\( x \)[/tex] approaches positive infinity ([tex]\( x \rightarrow \infty \)[/tex]):
- [tex]\( 3^x \)[/tex] becomes very large.
- [tex]\( g(x) = 3^x + 4 \)[/tex] will be dominated by the [tex]\( 3^x \)[/tex] term, and the function will go to infinity.
2. As [tex]\( x \)[/tex] approaches negative infinity ([tex]\( x \rightarrow -\infty \)[/tex]):
- [tex]\( 3^x \)[/tex] approaches 0 because any number raised to a negative power decreases towards zero.
- So, [tex]\( g(x) = 3^x + 4 \)[/tex] approaches [tex]\( 0 + 4 \)[/tex].
Hence, as [tex]\( x \)[/tex] approaches negative infinity, [tex]\( g(x) \)[/tex] approaches 4. Thus, the horizontal asymptote is:
[tex]\[ y = 4 \][/tex]
### Conclusion
The [tex]$y$[/tex]-intercept is [tex]\( (0, 5) \)[/tex] and the horizontal asymptote is [tex]\( y = 4 \)[/tex].
Thus, the correct multiple-choice answer is:
[tex]\[ (0, 5); y = 4 \][/tex]
