Question

function
exponential function:
f(x)=a·b^x
(use a = 1 and b = 2 to fill out this chart)
f(x)=1(2)^x
left end behavior (use limit notation):
horizontal asymptote(s): (write in the form of y = __ )
sketch of graph
right end behavior (use limit notation):
vertical asymptote(s): (write in the form of x = __ )
write using interval notation:
intervals of increasing:
intervals of decreasing:
write using interval notation
domain:
write using interval notation
range

Explanation:

Step1: Analyze left - end behavior

As \(x\to-\infty\), for \(y = 2^{x}\), \(\lim_{x\to-\infty}2^{x}=0\).

Step2: Analyze right - end behavior

As \(x\to+\infty\), for \(y = 2^{x}\), \(\lim_{x\to+\infty}2^{x}=+\infty\).

Step3: Determine horizontal asymptote

Since \(\lim_{x\to-\infty}2^{x}=0\), the horizontal asymptote is \(y = 0\).

Step4: Check for vertical asymptote

The function \(y = 2^{x}\) has no vertical asymptote as there are no values of \(x\) for which the function is undefined.

Step5: Find intervals of increasing/decreasing

The derivative of \(y = 2^{x}\) is \(y'=2^{x}\ln(2)>0\) for all \(x\in R\), so the function is increasing on \((-\infty,\infty)\) and decreasing on \(\varnothing\).

Step6: Determine domain and range

The domain of \(y = 2^{x}\) is all real numbers, written as \((-\infty,\infty)\) in interval notation. Since \(2^{x}>0\) for all \(x\in R\), the range is \((0,\infty)\).

Answer:

Property	Value
Right End Behavior	\(\lim_{x\to+\infty}2^{x}=+\infty\)
Horizontal Asymptote	\(y = 0\)
Vertical Asymptote	None
Intervals of Increasing	\((-\infty,\infty)\)
Intervals of Decreasing	\(\varnothing\)
Domain	\((-\infty,\infty)\)
Range	\((0,\infty)\)