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

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

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