Q. 57 - 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.

$f\left(x\right)=5-e^{-x}$

Step-by-Step Solution

Answer

Domain: All real numbers.

Range: $\left\{y|y<5\right\} or \left(-\infty ,5\right)$

Horizontal asymptote: $y=5$

The graph of the function is given below,

Step 1. Given information.

Consider the given function,

$f\left(x\right)=5-e^{-x}$

First, plot the graph of $y=e^x$, then multiply $-1$ and reflect about x-axis. Also replace x with $-x$ and reflect about y-axis.

Then add $5$ and shift up by $5units$,

Step 2. Determine the characteristics of the function.

Consider the given function,

$f\left(x\right)=5-e^{-x}$

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

The range is $\left\{y|y<5\right\} or \left(-\infty ,5\right)$.

Horizontal asymptote is $y=5$.