Question

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

(b) Find the points of intersection of the graphs of f and g by solving $f\left(x\right)=g\left(x\right)$Round answers to three decimal places. Label any intersection points on the graph drawn in part (a).

(c) Based on the graph, solve $f\left(x\right)>g\left(x\right)$.

Short answer

a) The graph can be given as

b) The point of intersection is $(0.71.6.541)$

c)$f\left(x\right)>g\left(x\right)$when$x>0.71$

Step-by-step solution

Given information

We are given

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

Part a) Step 2: Graph f(x)

We use a table to graph the function

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

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

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

Part a) Step 2: Graph the functions 

The graph can be given as

Part b) Step 1: Solve for f(x)=g(x)

We get,

$$\begin{gathered} f\left(x\right)=g\left(x\right) \\ {3}^{x+1}=2^{x+2} \\ \ln \left(3^{x+1}\right)=\ln \left(2^{x+2}\right) \\ (x+1)\ln 3=(x+2)\ln 2 \\ x(\ln 3)+\ln 3=x(\ln 2)+2(\ln 2) \\ x(\ln 3)-x(\ln 2)=2(\ln 2)-\ln 3 \\ x=\frac{2(\ln 2)-\ln 3}{(\ln 3)-(\ln 2)} \\ x=\frac{\ln \left({\displaystyle \frac{4}{3}}\right)}{\ln {\displaystyle \frac{3}{2}}} \\ x=0.71 \end{gathered}$$

Now we find the corresponding function value

$$\begin{gathered} f\left(x\right)=3^{x+1} \\ f(0.71)=3^{0.71+1} \\ f(0.71)=6.541 \end{gathered}$$

Hence the point of intersection is$(0.71,6.541)$

Part c)  Step1: Solve f(x)>g(x)

From part b) We know that the two graphs intersect at$(0.71,6.541)$ and from the graph we can conclude that$f\left(x\right)>g\left(x\right)$for$x>0.71$