Certainly! Let's break down the steps to find the asymptote and [tex]\( y \)[/tex]-intercept of the function [tex]\( f(x) = 3^{x+1} - 2 \)[/tex].
1. Finding the Asymptote:
The function [tex]\( f(x) = 3^{x+1} - 2 \)[/tex] is an exponential function.
- As [tex]\( x \)[/tex] approaches infinity, [tex]\( 3^{x+1} \)[/tex] grows very large.
- The term [tex]\(-2\)[/tex] becomes negligible in comparison to [tex]\( 3^{x+1} \)[/tex] as [tex]\( x \)[/tex] becomes very large.
- Therefore, the function approaches [tex]\( 3^{x+1} - 2 \)[/tex] but never actually reaches a horizontal line that would be considered the asymptote.
For large positive values of [tex]\( x \)[/tex], the function tends towards a line which is offset by [tex]\( -2 \)[/tex]. Thus, the horizontal asymptote of the function is at [tex]\( y = -2 \)[/tex].
2. Finding the [tex]\( y \)[/tex]-Intercept:
The [tex]\( y \)[/tex]-intercept occurs where the function intersects the [tex]\( y \)[/tex]-axis. This happens when [tex]\( x = 0 \)[/tex].
- Plug [tex]\( x = 0 \)[/tex] into the function:
[tex]\[
f(0) = 3^{0+1} - 2 = 3^1 - 2 = 3 - 2 = 1
\][/tex]
Therefore, the [tex]\( y \)[/tex]-intercept is [tex]\( 1 \)[/tex].
To summarize:
- The asymptote of the function [tex]\( f(x) = 3^{x+1} - 2 \)[/tex] is [tex]\( y = -2 \)[/tex].
- The [tex]\( y \)[/tex]-intercept is [tex]\( 1 \)[/tex].
Now let's fill in the blanks in the statement with the correct values:
The asymptote of the function [tex]\( f(x) = 3^{x+1} - 2 \)[/tex] is [tex]\( y = -2 \)[/tex]. Its [tex]\( y \)[/tex]-intercept is [tex]\( 1 \)[/tex].
