Answer:
Please check the attached graph.
From the graph, it is clear that option B is the correct option.
Step-by-step explanation:
Given the function
![g\left(x\right)=\:\frac{3}{2}\:\left(\frac{2}{3}\right)^x](https://tex.z-dn.net/?f=g%5Cleft%28x%5Cright%29%3D%5C%3A%5Cfrac%7B3%7D%7B2%7D%5C%3A%5Cleft%28%5Cfrac%7B2%7D%7B3%7D%5Cright%29%5Ex)
Determining the y-intercept
We know that the value of the y-intercept can be determined by setting x = 0, and determining the corresponding value of y.
so
substituting x = 0 in the fuction
![y=\:\frac{3}{2}\:\left(\frac{2}{3}\right)^x](https://tex.z-dn.net/?f=y%3D%5C%3A%5Cfrac%7B3%7D%7B2%7D%5C%3A%5Cleft%28%5Cfrac%7B2%7D%7B3%7D%5Cright%29%5Ex)
![y=\:\frac{3}{2}\:\left(\frac{2}{3}\right)^0](https://tex.z-dn.net/?f=y%3D%5C%3A%5Cfrac%7B3%7D%7B2%7D%5C%3A%5Cleft%28%5Cfrac%7B2%7D%7B3%7D%5Cright%29%5E0)
Apply rule: ![a^0=1,\:a\ne \:0](https://tex.z-dn.net/?f=a%5E0%3D1%2C%5C%3Aa%5Cne%20%5C%3A0)
![y=1\cdot \frac{3}{2}](https://tex.z-dn.net/?f=y%3D1%5Ccdot%20%5Cfrac%7B3%7D%7B2%7D)
![y=\frac{3}{2}](https://tex.z-dn.net/?f=y%3D%5Cfrac%7B3%7D%7B2%7D)
![y = 1.5](https://tex.z-dn.net/?f=y%20%3D%201.5)
Therefore, the point representing the y-intercept is:
Determining the x-intercept
We know that the value of the x-intercept can be determined by setting y = 0, and determining the corresponding value of x.
so
substituting y = 0 in the function
![0=\frac{3}{2}\left(\frac{2}{3}\right)^x](https://tex.z-dn.net/?f=0%3D%5Cfrac%7B3%7D%7B2%7D%5Cleft%28%5Cfrac%7B2%7D%7B3%7D%5Cright%29%5Ex)
Using the zero factor principle
if ab=0, then a=0 or b=0 (or both a=0 and b=0)
![\left(\frac{2}{3}\right)^x=0](https://tex.z-dn.net/?f=%5Cleft%28%5Cfrac%7B2%7D%7B3%7D%5Cright%29%5Ex%3D0)
We know that
can not be zero or negative for x ∈ R
Thus, NONE represents the x-intercept.
Please check the attached graph.
From the graph, it is clear that option B is the correct option.