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.

$\mathit{g}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathbf{1}\mathbf{-}{\mathbf{3}}^{\mathbf{-}\mathbf{x}}$

(Figure – 2)

Short answer

The graph is:

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

The range is $(-\infty ,1)$.

The horizontal asymptote is $(-\infty ,1)$.

Step-by-step solution

Step-1 – Given

The given function is $g\left(x\right)=1-3^{-x}$.

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 $g\left(x\right)=1-3^{-x}$is $y=3^{-x}={\left(\frac{1}{3}\right)}^x$.

To get the graph of $g\left(x\right)=1-3^{-x}$from $y={\left(\frac{1}{3}\right)}^x$we’ll reflect the function $y={\left(\frac{1}{3}\right)}^x$ over the x-axis. Then we move it 1 unit up.

So, the graph will be:

From the graph, we see the possible values of x are all real numbers. So, the domain is $(-\infty ,\infty )$.

The y-values are from$-\infty$to 1. So, the range is $(-\infty ,1)$.

And the horizontal asymptote of the graph is $y=1$.