Question

(a) Graph$f\left(x\right)=2^x$and $g\left(x\right)=12$on the same Cartesian plane.

(b) Shade the region bounded by the y-axis, $f\left(x\right)=2^x$, and $g\left(x\right)=12$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).

Short answer

a) The graph of the function can be given as

b) The region bounded by the function is

c) The point of intersection is$(3.585,12)$

Step-by-step solution

Given information

We are given

$$\begin{gathered} f\left(x\right)=2^x \\ g\left(x\right)=12 \end{gathered}$$

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

The table can be given as

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

The line y=12 is a line parallel to x-axis

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

The graph can be given as

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

The region in between the function is the region bounded by the function

It can be given as

Part c) Step 1: Find the point of intersection

At the point of intersection

$$\begin{gathered} f\left(x\right)=g\left(x\right) \\ {2}^x=12 \\ x(\ln 2)=\ln \left(12\right) \\ x=\frac{\ln 12}{\ln 2} \\ x=3.585 \end{gathered}$$

The corresponding function value is

$$\begin{gathered} f\left(x\right)=2^x \\ f(3.585)=2^{3.585} \\ f(3.585)=12 \end{gathered}$$

The point of intersection is$(3.585,12)$