Question

Graphing Exponential Functions Graph the function, not by plotting points, but by starting from the graphs in Figure 2.

State the domain, range, and asymptote.

$\mathit{h}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathbf{4}\mathbf{+}{\mathbf{\left(}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{\right)}}^{\mathbf{x}}$

(Figure – 2)

Short answer

The graph is:

The domain is $(-\infty ,\infty )$.

The range is $(4,\infty )$.

The horizontal asymptote is $y=4$.

Step-by-step solution

Step-1 – Given 

The given function is $h\left(x\right)=4+{\left(\frac{1}{2}\right)}^x$.

Step-2 – To determine

We have to sketch a graph for the given function using figure 2. Then we have to state its domain, range and asymptote.

Step-3 – Calculation

The parent function of $h\left(x\right)=4+{\left(\frac{1}{2}\right)}^x$is $y={\left(\frac{1}{2}\right)}^x$.

To get the graph of $h\left(x\right)=4+{\left(\frac{1}{2}\right)}^x$from $y={\left(\frac{1}{2}\right)}^x$we’ll move the function $y={\left(\frac{1}{2}\right)}^x$by 4 units up.

So, the graph will be:

From the graph, we see the possible values ofx are all real numbers. So, the domain is.

The y-values are from 4 to $\infty$. So, the range is $(4,\infty )$.

And the horizontal asymptote of the graph is $y=4$.