Answer:
Correct answer: C. f(x) = 2(3)x + 4
(please note we changed the expression of the function to exponential form which we believe is the correct form of the question)
Step-by-step explanation:
Horizontal Asymptote
The graph of a function is said to have a horizontal asymptote at y=a if one or both the following limits exist
[tex]\lim\limits_{x \rightarrow \infty}f(x)=a[/tex]
[tex]\lim\limits_{x \rightarrow -\infty}f(x)=a[/tex]
The horizontal asymptotes are horizontal lines to which the function tends when x increases or decreases without limits.
Let's analyze each one of the options provided:
A. f(x) = 2x -4
[tex]\lim\limits_{x \rightarrow \infty}(2x-4)=+\infty[/tex]
[tex]\lim\limits_{x \rightarrow -\infty}(2x-4)=-\infty[/tex]
No horizontal asymptote
B. f(x) = -3x + 4
[tex]\lim\limits_{x \rightarrow \infty}(-3x+4)=-\infty[/tex]
[tex]\lim\limits_{x \rightarrow -\infty}(-3x+4)=\infty[/tex]
No horizontal asymptote
C. f(x) = 2\cdot 3^x + 4
[tex]\lim\limits_{x \rightarrow \infty}(2\cdot 3^x + 4)=2\cdot 3^\infty + 4=\infty[/tex]
[tex]\lim\limits_{x \rightarrow \infty}(2\cdot 3^x + 4)=2\cdot 3^{-\infty} + 4=0+4=4[/tex]
This function has a horizontal asymptote at y=4
D. 3\cdot 2^x -4
[tex]\lim\limits_{x \rightarrow \infty}(3\cdot 2^x - 4)=3\cdot 2^\infty - 4=\infty[/tex]
[tex]\lim\limits_{x \rightarrow \infty}(3\cdot 2^x- 4)=3\cdot 2^{-\infty} - 4=0-4=-4[/tex]
This function has a horizontal asymptote at y=-4
Correct answer: C.
