Question

What is the shape of the graph of the function?
g(x)= 3/2 (2/3) ^x

Answer 1

Please check the attached graph.

From the graph, it is clear that option B is the correct option.

Explanation:

Given the function

`g\left(x\right)=\:(3)/(2)\:\left((2)/(3)\right)^x`

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=\:(3)/(2)\:\left((2)/(3)\right)^x`

`y=\:(3)/(2)\:\left((2)/(3)\right)^0`

Apply rule:
`a^0=1,\:a\\e \:0`

`y=1\cdot (3)/(2)`

`y=(3)/(2)`

`y = 1.5`

Therefore, the point representing the y-intercept is:

• (0, 1.5)

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=(3)/(2)\left((2)/(3)\right)^x`

Using the zero factor principle

if ab=0, then a=0 or b=0 (or both a=0 and b=0)

`\left((2)/(3)\right)^x=0`

We know that
`a^(f\left(x\right))` 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.