Answer:
<em>Correct answer: C. f(x) = 2(3)x + 4</em>
<u><em>(please note we changed the expression of the function to exponential form which we believe is the correct form of the question)</em></u>
Step-by-step explanation:
<u>Horizontal Asymptote</u>
The graph of a function is said to have a horizontal asymptote at y=a if one or both the following limits exist
![\lim\limits_{x \rightarrow \infty}f(x)=a](https://tex.z-dn.net/?f=%5Clim%5Climits_%7Bx%20%5Crightarrow%20%5Cinfty%7Df%28x%29%3Da)
![\lim\limits_{x \rightarrow -\infty}f(x)=a](https://tex.z-dn.net/?f=%5Clim%5Climits_%7Bx%20%5Crightarrow%20-%5Cinfty%7Df%28x%29%3Da)
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
![\lim\limits_{x \rightarrow \infty}(2x-4)=+\infty](https://tex.z-dn.net/?f=%5Clim%5Climits_%7Bx%20%5Crightarrow%20%5Cinfty%7D%282x-4%29%3D%2B%5Cinfty)
![\lim\limits_{x \rightarrow -\infty}(2x-4)=-\infty](https://tex.z-dn.net/?f=%5Clim%5Climits_%7Bx%20%5Crightarrow%20-%5Cinfty%7D%282x-4%29%3D-%5Cinfty)
No horizontal asymptote
B. f(x) = -3x + 4
![\lim\limits_{x \rightarrow \infty}(-3x+4)=-\infty](https://tex.z-dn.net/?f=%5Clim%5Climits_%7Bx%20%5Crightarrow%20%5Cinfty%7D%28-3x%2B4%29%3D-%5Cinfty)
![\lim\limits_{x \rightarrow -\infty}(-3x+4)=\infty](https://tex.z-dn.net/?f=%5Clim%5Climits_%7Bx%20%5Crightarrow%20-%5Cinfty%7D%28-3x%2B4%29%3D%5Cinfty)
No horizontal asymptote
C. f(x) = 2\cdot 3^x + 4
![\lim\limits_{x \rightarrow \infty}(2\cdot 3^x + 4)=2\cdot 3^\infty + 4=\infty](https://tex.z-dn.net/?f=%5Clim%5Climits_%7Bx%20%5Crightarrow%20%5Cinfty%7D%282%5Ccdot%203%5Ex%20%2B%204%29%3D2%5Ccdot%203%5E%5Cinfty%20%2B%204%3D%5Cinfty)
![\lim\limits_{x \rightarrow \infty}(2\cdot 3^x + 4)=2\cdot 3^{-\infty} + 4=0+4=4](https://tex.z-dn.net/?f=%5Clim%5Climits_%7Bx%20%5Crightarrow%20%5Cinfty%7D%282%5Ccdot%203%5Ex%20%2B%204%29%3D2%5Ccdot%203%5E%7B-%5Cinfty%7D%20%2B%204%3D0%2B4%3D4)
This function has a horizontal asymptote at y=4
D. 3\cdot 2^x -4
![\lim\limits_{x \rightarrow \infty}(3\cdot 2^x - 4)=3\cdot 2^\infty - 4=\infty](https://tex.z-dn.net/?f=%5Clim%5Climits_%7Bx%20%5Crightarrow%20%5Cinfty%7D%283%5Ccdot%202%5Ex%20-%204%29%3D3%5Ccdot%202%5E%5Cinfty%20-%204%3D%5Cinfty)
![\lim\limits_{x \rightarrow \infty}(3\cdot 2^x- 4)=3\cdot 2^{-\infty} - 4=0-4=-4](https://tex.z-dn.net/?f=%5Clim%5Climits_%7Bx%20%5Crightarrow%20%5Cinfty%7D%283%5Ccdot%202%5Ex-%204%29%3D3%5Ccdot%202%5E%7B-%5Cinfty%7D%20-%204%3D0-4%3D-4)
This function has a horizontal asymptote at y=-4
Correct answer: C.