Answer:
[tex]\textsf{domain:}\quad-1\leq x\leq 3[/tex]
[tex]\textsf{range:}\quad\dfrac{16}{3}\leq y\leq 27[/tex]
Step-by-step explanation:
Information
Domain of a function: set of all possible input values (x-values)
Range of a function: set of all possible output values (y-values)
Closed circle: less than or equal to and greater than or equal to (≤ or ≥).
Open circle: less than or greater than (< or >)
Domain
There is a closed dot at x = -1 and x = 3
Therefore, the domain is -1 ≤ x ≤ 3
Range
To calculate the exact range, we need to figure out the equation of the function (as it is difficult to read the exact value of y at x = -1).
From inspection of the graph, we can identify the following ordered pairs:
(1, 12) (2, 18) (3, 27)
General form of an exponential equation: [tex]y=ab^x[/tex]
Inputting the first two ordered pairs into the exponential equation:
[tex]\implies ab^1=12[/tex]
[tex]\implies ab^2=18[/tex]
Dividing the equations to find b:
[tex]\implies \dfrac{ab^2}{ab^1}=\dfrac{18}{12}[/tex]
[tex]\implies b=\dfrac32[/tex]
Inputting b into the first equation to find a:
[tex]\implies \dfrac32a=12[/tex]
[tex]\implies a=8[/tex]
Therefore, the equation of the function is: [tex]y=8\left(\dfrac32\right)^x[/tex]
Input [tex]x = -1[/tex] into the equation to find y:
[tex]\implies 8\left(\dfrac32\right)^{-1}=\dfrac{16}{3}[/tex]
Therefore, the range is [tex]\dfrac{16}{3}\leq y\leq 27[/tex]
