We have the function y=8^x. This is an exponential function.
The domain is the set of values of x for which the functions is defined. We have no restriction for the values of x because y is defined for all real values of x, so the domain is:
[tex]D=\mleft\lbrace x\mright|\text{ all real numbers}\}[/tex]The range is the set of values of y for the domain for which the function is defined.
We know that y will not take negative values: any value of x as input will give us a positive value of y, so the range can be defined as:
[tex]R\colon\mleft\lbrace y\mright|y>0\}[/tex]The y-intercept is the value of y when the function intersects the y-axis. This happens when x=0. Then, the y-intercept is:
[tex]y(0)=8^0=1[/tex]We know that there are no vertical assymptotes, because there is no singularity in the function (the function is defined for all real numbers), so we will look for horizontal assymptotes:
[tex]\begin{gathered} 1)\lim _{x\longrightarrow\infty}8^x=\infty \\ 2)\lim _{x\longrightarrow-\infty}8^x=\lim _{x\longrightarrow}\frac{1}{8^{-x}}=\frac{1}{\infty}=0 \end{gathered}[/tex]As we have a finite number for the second limit, we have an assymptote at y=0.
We can see if the function is increasing or decreasing by comparing:
[tex]\begin{gathered} a>b \\ f(a)=8^a \\ f(b)=8^b \\ \frac{f(b)}{f(a)}=\frac{8^b}{8^a}=8^{b-a}>1\Rightarrow f(b)>f(a) \end{gathered}[/tex]As the function value increases with the increase of x, we know that the function is increasing for all the values of the domain.
We can graph the function as:
Answer:
Domain: D: {x | all real numbers}
Range: R: {y | y>0}
Y-intercept: y(0) = 1
Assymptote: y=0
The function is increasing for all values of x.
