Question

A) Graph the function: f(x) = 2^x+^4B)domain of the function?C) range of the function ?D) equation of the asymptote?E) y-intercept of the graph?

Answer 1

Given:

`f\mleft(x\mright)=2^(x+4)`

A) Graph the function:

Let us find the intercepts.

When x=0, we get

`\begin{gathered} f\mleft(0\mright)=2^4 \\ f(0)=16 \end{gathered}`

Since it is an exponential function.

So, the graph is,

B) To find the domain:

According to the graph,

The domain is,

`(-\infty,\infty)`

C) To find the range:

According to the graph,

The range is,

`(0,\infty)`

D) To find the asymptote:

The line y=L is a horizontal asymptote of the function y=f(x), if either

`\begin{gathered} \lim _(x\to\infty)f\mleft(x\mright)=L\text{ (or)} \\ \lim _(x\to-\infty)f\mleft(x\mright)=L \end{gathered}`

And L is finite.

Here,

`\begin{gathered} \lim _(x\to\infty)f(x)=2^(\infty+4) \\ =\infty \\ \lim _(x\to-\infty)f(x)=2^(-\infty+4) \\ =0 \end{gathered}`

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.