Q. 60 - Precalculus Enhanced with Graphing Utilities

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.

$f\left(x\right)=7-3e^{2x}$

Step-by-Step Solution

Answer

The graph of the function is given below,

Domain: All real numbers.

Range:$\left(-\infty ,7\right) or \left\{y|y<7\right\}$.

Horizontal asymptote: $y=7$

Step 1. Given information.

Consider the given function,

$f\left(x\right)=7-3e^{2x}$

First, plot the graph of $y=e^x$. Replace x with $2x$. Multiply by $-1$ and reflect about y-axis. Add $7$ and shift up by $7units$,

Step 2. Determine the characteristics of the function.

Consider the given function,

$f\left(x\right)=7-3e^{2x}$

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

As we can see that $x\to -\infty ,f\left(x\right)\to 7$.

Thus the range is $\left(-\infty ,7\right)$ or in the interval $\left\{y|y<7\right\}$.

Horizontal asymptote is  $y=7$.