To determine the range of the function [tex]\( g(x) = \frac{2}{3} x - 1 \)[/tex] given the domain [tex]\( (-\infty, 3] \)[/tex], we need to understand how the values of [tex]\( g(x) \)[/tex] vary as [tex]\( x \)[/tex] takes on values within this domain.
1. Understanding the Linear Function:
The function [tex]\( g(x) = \frac{2}{3} x - 1 \)[/tex] is a linear function. Linear functions produce straight lines when graphed and have the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
Here, [tex]\( m = \frac{2}{3} \)[/tex] and [tex]\( b = -1 \)[/tex].
2. Evaluating at the Boundary Conditions:
- As [tex]\( x \)[/tex] approaches [tex]\(-\infty\)[/tex]:
[tex]\[
g(x) = \frac{2}{3} x - 1 \Rightarrow \text{As } x \to -\infty, \; g(x) \to -\infty
\][/tex]
This means that as [tex]\( x \)[/tex] becomes arbitrarily large in the negative direction, [tex]\( g(x) \)[/tex] becomes arbitrarily large in the negative direction as well.
- At the upper limit of the domain, [tex]\( x = 3 \)[/tex]:
[tex]\[
g(3) = \frac{2}{3} \cdot 3 - 1 = 2 - 1 = 1
\][/tex]
This means that when [tex]\( x \)[/tex] reaches its maximum value of 3 within the domain, [tex]\( g(x) \)[/tex] reaches its maximum value of 1.
3. Determining the Range:
Given the behavior of [tex]\( g(x) \)[/tex] at the boundaries:
- As [tex]\( x \)[/tex] decreases without bound, [tex]\( g(x) \)[/tex] also decreases without bound, meaning [tex]\( g(x) \to -\infty \)[/tex].
- When [tex]\( x \)[/tex] is at its maximum value of 3, [tex]\( g(x) \)[/tex] reaches exactly 1.
From these observations, the range of [tex]\( g(x) \)[/tex] includes all values starting from [tex]\(-\infty\)[/tex] up to and including 1, because [tex]\( g(x) \)[/tex] can take any value in this interval as [tex]\( x \)[/tex] varies over its domain.
Therefore, the range of [tex]\( g(x) \)[/tex] given the domain [tex]\( (-\infty, 3] \)[/tex] is:
[tex]\[
\boxed{(-\infty, 1]}
\][/tex]
