[tex]\begin{gathered} \text{for }f\mleft(x\mright)=1/2^x\colon\text{ 4, 2, 1} \\ \text{for }f\mleft(x\mright)=2^x\text{ : 2, 4} \end{gathered}[/tex]
See explanation and graph below
Explanation:
For x less than or equal to zero, we would apply the function f(x) = (1/2)^x
For x greater than zero, we would apply the function f(x) = 2^x
when x = - 2 (less than 0)
This falls in the 1st function
[tex]\begin{gathered} f(-2)\text{ = (}\frac{1}{2})^{-2} \\ f(-2)=\frac{1}{(\frac{1}{2})^2}\text{ = 1}\times\frac{4}{1} \\ f(-2)=2^2\text{ = 4} \end{gathered}[/tex]
when x = -1 (less than 0)
This falls in the 1st function
[tex]\begin{gathered} f(-1)\text{ = (}\frac{1}{2})^{-1} \\ f(-1)\text{ = }\frac{1}{(\frac{1}{2})^1}\text{ = 2} \end{gathered}[/tex]
when x = 0 (equal to 0)
This falls in the 1st function
[tex]\begin{gathered} f(0)\text{ = (}\frac{1}{2})^0 \\ f(0)\text{ = 1} \end{gathered}[/tex]
when x = 1 (greater than 0)
This falls in the 2nd function
[tex]\begin{gathered} f(1)=2^1 \\ f(1)\text{ = 2} \end{gathered}[/tex]
when x = 2 (greater than 0)
THis falls in the 2nd function
[tex]\begin{gathered} f(2)\text{ = }2^2 \\ f(2)\text{ = 4} \end{gathered}[/tex]
Plotting the graph:
The end with the shaded dot reresent the function with equal to sign attached to the inequality [f(x) = (1/2)^x].
The end with the open dot represent the function without the equal to sign [f(x) = 2^x)
