Chapter 5.6, Problem 93AYU

(a) Graph $f\left(x\right)=2^{x+1}$ and $g\left(x\right)=2^{-x+2}$ on the same Cartesian plane.

(b) Shade the region bounded by the $y\text{-axis}$, $f\left(x\right)=2^{x+1}$ and $g\left(x\right)=2^{-x+2}$ on the graph drawn in part (a).

(c) Solve $f\left(x\right)=g\left(x\right)$ and label the point of intersection on the graph drawn in part (a).

To determine

To find:

a. Graph $f$ and $g$ on the same Cartesian plane.

Answer to Problem 93AYU

a.

Explanation of Solution

Given:

$f\left(x\right)=2^{x+1}$ and $g\left(x\right)=2^{-x+2}$

Calculation:

a.

To determine

To find:

b. Shade the region bounded by the $y\text{-axis}$, $f$ and $g$ on the graph drawn in part (a).

Answer to Problem 93AYU

b.

Explanation of Solution

Given:

$f\left(x\right)=2^{x+1}$ and $g\left(x\right)=2^{-x+2}$

Calculation:

b.

To determine

To find:

c. Solve $f\left(x\right)=g\left(x\right)$ and label the point of intersection on the graph drawn in part (a).

Answer to Problem 93AYU

c. $\left(\frac{1}{2},2\sqrt{2}\right)$

Explanation of Solution

Given:

$f\left(x\right)=2^{x+1}$ and $g\left(x\right)=2^{-x+2}$

Calculation:

c. $f\left(x\right)=g\left(x\right)$

$2^{x+1}=2^{-x+2}$

${\text{log}}_2{2}^{x+1}={\text{ log}}_2{2}^{-x+2}$

$x+1=-x+2$

$2x=1$

$x=\frac{1}{2}$

When $x=\frac{1}{2}$; $f\left(x\right)=2^{\frac{1}{2}+1}=2^{\frac{3}{2}}=\sqrt{8}=2\sqrt{2}$,

The point is $\left(\frac{1}{2},2\sqrt{2}\right)$.