Answer :
To find the asymptote and the y-intercept of the function [tex]\( f(x) = 3^{x+1} - 2 \)[/tex], we'll follow these steps:
### Finding the Asymptote
1. Understanding an Asymptote: An asymptote is a line that the graph of a function approaches but never touches.
2. Identify the Asymptote: Since [tex]\( 3^{x+1} \)[/tex] is an exponential function, as [tex]\( x \)[/tex] approaches negative infinity, [tex]\( 3^{x+1} \)[/tex] approaches 0. Thus, the function [tex]\( f(x) \)[/tex] will approach [tex]\( -2 \)[/tex], as the value -2 does not get influenced by the exponential term.
Therefore, the horizontal asymptote of the function [tex]\( f(x) = 3^{x+1} - 2 \)[/tex] is [tex]\( y = -2 \)[/tex].
### Finding the Y-intercept
1. Definition: The y-intercept of a function is the value of [tex]\( f(x) \)[/tex] when [tex]\( x = 0 \)[/tex].
2. Calculate [tex]\( f(0) \)[/tex]:
[tex]\[ f(0) = 3^{0+1} - 2 = 3^1 - 2 = 3 - 2 = 1 \][/tex]
Hence, the y-intercept of the function is [tex]\( (0, 1) \)[/tex].
### Conclusion
Now we select the correct answers for both dropdown menus:
1. The asymptote of the function [tex]\( f(x) = 3^{x+1} - 2 \)[/tex] is [tex]\( -2 \)[/tex].
2. Its y-intercept is [tex]\( 1 \)[/tex].
So, the completed sentence will be:
"The asymptote of the function [tex]\( f(x)=3^{x+1}-2 \)[/tex] is -2. Its y-intercept is 1."
### Finding the Asymptote
1. Understanding an Asymptote: An asymptote is a line that the graph of a function approaches but never touches.
2. Identify the Asymptote: Since [tex]\( 3^{x+1} \)[/tex] is an exponential function, as [tex]\( x \)[/tex] approaches negative infinity, [tex]\( 3^{x+1} \)[/tex] approaches 0. Thus, the function [tex]\( f(x) \)[/tex] will approach [tex]\( -2 \)[/tex], as the value -2 does not get influenced by the exponential term.
Therefore, the horizontal asymptote of the function [tex]\( f(x) = 3^{x+1} - 2 \)[/tex] is [tex]\( y = -2 \)[/tex].
### Finding the Y-intercept
1. Definition: The y-intercept of a function is the value of [tex]\( f(x) \)[/tex] when [tex]\( x = 0 \)[/tex].
2. Calculate [tex]\( f(0) \)[/tex]:
[tex]\[ f(0) = 3^{0+1} - 2 = 3^1 - 2 = 3 - 2 = 1 \][/tex]
Hence, the y-intercept of the function is [tex]\( (0, 1) \)[/tex].
### Conclusion
Now we select the correct answers for both dropdown menus:
1. The asymptote of the function [tex]\( f(x) = 3^{x+1} - 2 \)[/tex] is [tex]\( -2 \)[/tex].
2. Its y-intercept is [tex]\( 1 \)[/tex].
So, the completed sentence will be:
"The asymptote of the function [tex]\( f(x)=3^{x+1}-2 \)[/tex] is -2. Its y-intercept is 1."
