Given:
[tex]f\mleft(x\mright)=2^{x+4}[/tex]
A) Graph the function:
Let us find the intercepts.
When x=0, we get
[tex]\begin{gathered} f\mleft(0\mright)=2^4 \\ f(0)=16 \end{gathered}[/tex]
Since it is an exponential function.
So, the graph is,
B) To find the domain:
According to the graph,
The domain is,
[tex](-\infty,\infty)[/tex]
C) To find the range:
According to the graph,
The range is,
[tex](0,\infty)[/tex]
D) To find the asymptote:
The line y=L is a horizontal asymptote of the function y=f(x), if either
[tex]\begin{gathered} \lim _{x\to\infty}f\mleft(x\mright)=L\text{ (or)} \\ \lim _{x\to-\infty}f\mleft(x\mright)=L \end{gathered}[/tex]
And L is finite.
Here,
[tex]\begin{gathered} \lim _{x\to\infty}f(x)=2^{\infty+4} \\ =\infty \\ \lim _{x\to-\infty}f(x)=2^{-\infty+4} \\ =0 \end{gathered}[/tex]
Thus, the horizontal asymptote is y=0.
E) To find the y-intercept:
According to the graph,
When x=0, then f(x)=16.
So,
The y-intercept is 16.
