Question

A. What is the domain ?B. What is the range ?C. What is the x-intercept ?D. What is the y- intercept ?E. What is the asymptote ?

Answer 1

`\begin{gathered} Given \\ f(x)=2(3)^(x-1)+2 \end{gathered}`

Part A: Domain of the function

Since the given function is an exponential function, then the domain is

`\left(-\infty,\infty\right)`

Part B: Range of the function

Since there is a constant 2 in the function, the range of the function is

`(2,\infty)`

Part C: x-intercept

As the range starts at the horizontal asymptote of 2, then there is no x-intercept in the function

Part D: y-intercept

`\begin{gathered} f\mleft(x\mright)=2\left(3\right)^(x-1)+2 \\ f\mleft(0\mright)=2(3)^(0-1)+2 \\ f\mleft(0\mright)=2\cdot(1)/(3)+2 \\ f\mleft(0\mright)=(2)/(3)+2 \\ f\mleft(0\mright)=(8)/(3) \end{gathered}`

Therefore, the y-intercept is at (0,8/3).

Part E: Asymptote

Since the range starts at 2, the horizontal asymptote of the function is y = 2