Question

Mousio asked Feb 23, 2023

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2 Answers

Execjosh · Answer 1 · 2023-02-26T09:33:14+0000

Answer:

$\textsf{domain:}\quad-1\leq x\leq 3$

$\textsf{range:}\quad(16)/(3)\leq y\leq 27$

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:
y=ab^x

Inputting the first two ordered pairs into the exponential equation:

$\implies ab^1=12$

$\implies ab^2=18$

Dividing the equations to find b:

$\implies (ab^2)/(ab^1)=(18)/(12)$

$\implies b=\frac32$

Inputting b into the first equation to find a:

$\implies \frac32a=12$

$\implies a=8$

Therefore, the equation of the function is:
$y=8\left(\frac32\right)^x$

Input
x = -1 into the equation to find y:

$\implies 8\left(\frac32\right)^(-1)=(16)/(3)$

Therefore, the range is
$(16)/(3)\leq y\leq 27$

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