Q. 59 - 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.

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

Step-by-Step Solution

Answer

The graph of the function is given below,

Domain: All real numbers.

Range: $\left\{y|y<2\right\} or \left(-\infty ,2\right)$

Horizontal asymptote: $y=2$

Step 1. Given information.

Consider the given function,

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

First, plot the graph of $y=e^x$. Replace x with $-x/2$ and reflect about x-axis. Multiply with $-1$ and reflect about y-axis.

Then add $2$ and shift up by $2units$,

Step 2. Determine the characteristics of the function.

Consider the given function,

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

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

The range is $\left\{y|y<2\right\}$ or in the interval $\left(-\infty ,2\right)$.

Horizontal asymptote is $y=2$.