Question

Begin with the graph of $y=e^x$and use transformation to graph the function. Determine the domain, range and horizontal asymptote of the function.

$f\left(x\right)=9-3e^{-x}$

Short answer

The graph of the function is given below,

Domain: All real numbers.

Range: $\left(-\infty ,9\right)$

Horizontal asymptote:$y=9$

Step-by-step solution

Step 1. Given information.

Consider the given function,

$f\left(x\right)=9-3e^{-x}$

First, plot the graph of $y=e^x$, then replace x with $-x$ and reflect about y-axis.

Multiply by $-3$ and reflect about x-axis.

Then add $9$and shift up by $9units$,

Step 2. Determine the characteristics of the function.

Consider the given function,

$f\left(x\right)=9-3e^{-x}$

From the graph, the domain of the function is the set of all real numbers.

As we can see that$x\to -\infty ,f\left(x\right)\to 9$.

The range is $\left(-\infty ,9\right)$.

Horizontal asymptote is$y=9$.