The function provided is $f(x) = -\frac{3}{2}x + 3$, which is a linear function based on the equation. Let's break down the domain and range of this function:

1. Function Family:

• The function belongs to the linear function family, as it represents a straight line on the graph.

2. Domain:

• The domain of a linear function is all real numbers, since there are no restrictions on $x$. This means the function is defined for all possible values of $x$.

  So, the domain is: $\text{Domain} = (-\infty, \infty)$

3. Range:

• The range of a linear function is also all real numbers, because as $x$ moves from $-\infty$ to $\infty$, $f(x)$ will take all possible values.

  Thus, the range is: $\text{Range} = (-\infty, \infty)$

However, the text and graph provided in your image are discussing specific intervals, possibly focusing on restrictions based on the context of the problem (e.g., the line crossing certain points on the graph). The solution seems to mention conditions like $x < -\frac{3}{2}$ or $x > \frac{3}{2}$, which may imply further details or constraints on where the function applies in that particular case.

Let me know if you'd like a detailed explanation of the intervals, or if you have any other questions!

Related Questions:

1. What are the properties of a linear function's graph?
2. How do you find the slope and y-intercept of a line given its equation?
3. How does the slope affect the range and behavior of a linear function?
4. Why do linear functions have an unrestricted domain and range?
5. What changes in the graph of a linear function when the slope or y-intercept changes?

Tip: To quickly determine the domain and range of any linear function, remember that its graph extends infinitely in both the $x$- and $y$-directions unless restrictions are explicitly given.