given the function below, fill in the table of values, use the table of values to graph the function, and…
given the function below, fill in the table of values, use the table of values to graph the function, and then identify the functions domain and range. \\( y = \\left( \\frac{1}{2} \\right)^x \\) \\( \\begin{array}{|c|c|c|c|c|c|c|} \\hline x & -3 & -2 & -1 & 0 & 1 & 2 \\\\ \\hline y = \\left( \\frac{1}{2} \\right)^x & \\boxed{} & & & & & \\\\ \\hline \\end{array} \\)
Answer
Explanation:
Step1: Calculate y for x=-3
Substitute $x=-3$ into $y=\left(\frac{1}{2}\right)^x$. Using the property $a^{-n}=\frac{1}{a^n}$, we get $\left(\frac{1}{2}\right)^{-3}=2^3=8$.
Step2: Calculate y for x=-2
Substitute $x=-2$ into $y=\left(\frac{1}{2}\right)^x$. $\left(\frac{1}{2}\right)^{-2}=2^2=4$.
Step3: Calculate y for x=-1
Substitute $x=-1$ into $y=\left(\frac{1}{2}\right)^x$. $\left(\frac{1}{2}\right)^{-1}=2^1=2$.
Step4: Calculate y for x=0
Substitute $x=0$ into $y=\left(\frac{1}{2}\right)^x$. Any non-zero number to the 0 power is 1, so $\left(\frac{1}{2}\right)^0=1$.
Step5: Calculate y for x=1
Substitute $x=1$ into $y=\left(\frac{1}{2}\right)^x$. $\left(\frac{1}{2}\right)^1=\frac{1}{2}$.
Step6: Calculate y for x=2
Substitute $x=2$ into $y=\left(\frac{1}{2}\right)^x$. $\left(\frac{1}{2}\right)^2=\frac{1}{4}$.
Step7: Identify domain
For exponential functions, $x$ can be any real number, so domain is all real numbers.
Step8: Identify range
The output $y=\left(\frac{1}{2}\right)^x$ is always positive and never equals 0, so range is all positive real numbers.
Answer:
Completed Table:
| $x$ | $y=\left(\frac{1}{2}\right)^x$ |
|---|---|
| $-3$ | $8$ |
| $-2$ | $4$ |
| $-1$ | $2$ |
| $0$ | $1$ |
| $1$ | $\frac{1}{2}$ |
| $2$ | $\frac{1}{4}$ |