graph the exponential function g(x)=3^(x + 1)...
graph the exponential function g(x)=3^(x + 1). to do this, plot two points on the graph of the function, and also draw the asymptote. then click on the graph - a - function button. additionally, give the domain and range of the function using interval notation.
Answer
# Explanation:
## Step1: Find two points on the function
Let \(x = - 1\), then \(g(-1)=3^{-1 + 1}=3^{0}=1\). Let \(x = 0\), then \(g(0)=3^{0 + 1}=3^{1}=3\). So two points are \((-1,1)\) and \((0,3)\).
## Step2: Determine the asymptote
For an exponential - function of the form \(y = a\cdot b^{x + h}+k\), in the function \(g(x)=3^{x + 1}\), which can be written as \(y = 3\cdot3^{x}\), the horizontal asymptote is \(y = 0\) since there is no vertical shift (\(k = 0\)).
## Step3: Find the domain
The domain of an exponential function \(y = b^{x}\) (in this case \(y = 3^{x+1}\)) is all real numbers. In interval notation, the domain is \((-\infty,\infty)\).
## Step4: Find the range
Since the horizontal asymptote is \(y = 0\) and the function \(y = 3^{x+1}\) is always positive (because \(b = 3>0\)), the range is \((0,\infty)\).
# Answer:
Two points: \((-1,1)\) and \((0,3)\); Asymptote: \(y = 0\); Domain: \((-\infty,\infty)\); Range: \((0,\infty)\)