Question

(a) Graph $f\left(x\right)=3^{-x+1}$and $g\left(x\right)=3^{x-2}$on the same Cartesian plane.

(b) Shade the region bounded by the y-axis, $f\left(x\right)=3^{-x+1}$and $g\left(x\right)=3^{x-2}$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

Step-by-step solution

Given information

We are given functions

$$\begin{gathered} f\left(x\right)=3^{-x+1} \\ g\left(x\right)=3^{x-2} \end{gathered}$$

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

The table for the function $f\left(x\right)=3^{-x+1}$

x	$f\left(x\right)=3^{-x+1}$
-2	$\frac{1}{3}$
-1	1
0	3
1	9
2	27

Similarly the table for the function $g\left(x\right)=3^{x-2}$can be given as

x	$g\left(x\right)=3^{x-2}$
0	$\frac{1}{9}$
1	$\frac{1}{3}$
2	1
3	3
4	9

Part a) Step 2: Plot the graph of the function

The graph of the function can be given as

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

We get,

Part c) Step 1: Find the point of intersection

As the bases are same

$$\begin{gathered} 3^{-x+1}=3^{x-2} \\ -x+1=x-2 \\ 2x=3 \\ x=1.5 \end{gathered}$$

The corresponding function value is

$$\begin{gathered} f\left(x\right)=3^{-x+1} \\ f(-0.5)=3^{-1.5+1} \\ f(-0.5)=0.577 \end{gathered}$$

The point of intersection is$(-0.5,5.196)$

part c) Step 2:We shoe the point of intersection on the graph

We have,