Question

exponential function characteristics
what are the features of the graph?
graph ( f(x)=2(3)^{x}+1 )
you may use the desmos graphing calculator.
horizontal asymptote
( y= )
the function is on its domain
( (-infty,infty) )
the left end behavior as ( x\to-infty )
the right end behavior as ( x\to+infty )

Explanation:

Step1: Recall the general form of an exponential function

The general form of an exponential function is \(y = a(b)^x + c\). For the function \(f(x)=2(3)^x + 1\), where \(a = 2\), \(b=3\), \(c = 1\). The horizontal asymptote of the function \(y=a(b)^x + c\) is \(y = c\) when \(|b|>1\) or \(0<|b|<1\).

Step2: Determine the horizontal asymptote

Since \(c = 1\), the horizontal asymptote is \(y = 1\).

Step3: Analyze the behavior of the function

For an exponential function \(y=a(b)^x + c\) with \(a>0\) and \(b > 1\):

• The function \(y = 2(3)^x+1\) is increasing on its domain \((-\infty,\infty)\) because the base \(b = 3>1\) and \(a=2>0\).
• As \(x\to-\infty\), \(3^x=\frac{1}{3^{-x}}\to0\). So \(y=2(3)^x + 1\to1\).
• As \(x\to+\infty\), \(3^x\to+\infty\). So \(y=2(3)^x + 1\to+\infty\).

Answer:

Horizontal Asymptote \(y = 1\); The function is increasing on its domain \((-\infty,\infty)\); The left - end behavior as \(x\to-\infty\) is \(y\to1\); The right - end behavior as \(x\to+\infty\) is \(y\to+\infty\)