Question

(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-axis, $f\left(x\right)=2^{x+1}$and $g\left(x\right)=2^{-x+2}$on the graph draw 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).

Short answer

a) The graph of the function can be given as

2) the region bounded by the function is

c) The point of intersection is$(0.5,2.828)$

Step-by-step solution

Given information

We are given

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

Part a) Step 1: We create table to plot the graph

The table for $f\left(x\right)=2^{x+1}$

x	$f\left(x\right)=2^{x+1}$
-2	0.5
-1	1
0	2
1	4
2	8

Similarly the table for $g\left(x\right)=2^{-x+2}$

x	$g\left(x\right)=2^{-x+2}$
-3	32
-2	16
-1	8
0	4
1	2

Part a) Step 2: Plot the graph

We have,

Part b) Step 1: Shade the region bounded by the function

We get,

Part c) Step 1: Find the point of intersection

At point of intersection,

$$\begin{gathered} f\left(x\right)=g\left(x\right) \\ {2}^{x+1}=2^{-x+2} \\ x+1=-x+2 \\ 2x=1 \\ x=\frac{1}{2} \end{gathered}$$

The corresponding function value is

$$\begin{gathered} f\left(x\right)=2^{x+1} \\ f(0.5)=2^{0.5+1} \\ f(0.5)=2^{1.5} \\ f(0.5)=2.828 \end{gathered}$$

The point of intersection is$(0.5,2.828)$

Part c) Step 3: We show the point of intersection on the graph

We get,