Chapter 3, Problem 15RE

Solution

To graph: Graph the function $f\left(x\right)=e{}^{3–x}+1$ , and analyze it for domain, range, continuity, increasing or decreasing behavior, symmetry, boundedness, extrema, asymptotes, and end behavior.

Given information: The function $f\left(x\right)=e{}^{3–x}+1$ .

Graph:

The given function is- $f\left(x\right)=e{}^{3–x}+1$

Rewrite the function with the property, $e^{a-b}=\frac{e^a}{e^b}$ .

  $\begin{array}{l}f\left(x\right)=e^{3-x}+1\\ f\left(x\right)=\frac{e^3}{e^x}+1\end{array}$

  

Interpretation:

Domain: All real $x$ .

Range: All real $y>1$ .

Continuity: Function is continuous on its domain.

Increasing or decreasing behavior: Function is decreasing on its domain.

Symmetry: Function is not containing any symmetry.

Bounded: Function is bounded below at $y=1$ .

Extrema: Function has no one extrema.

Asymptotes: Function has one horizontal asymptote at $y=1$ .

End behavior: $\underset{x\to \infty }{\lim }f\left(x\right)=1$ and $\underset{x\to -\infty }{\lim }f\left(x\right)=\infty$ .