The given problem asks to graph the linear function and to determine its domain and range. The function provided is:

$f(x) = -5x + 3$

1. Graphing the Linear Function

The equation $f(x) = -5x + 3$ is in the slope-intercept form, $y = mx + b$, where:

• $m = -5$ is the slope of the line.
• $b = 3$ is the y-intercept (where the line crosses the y-axis).

Steps to graph:

1. Start at the y-intercept $(0, 3)$.
2. Use the slope $-5$ (which means for every 1 unit you move to the right, you move 5 units down) to plot additional points.
3. Connect the points to form the line.

2. Domain and Range

• Domain: Since this is a linear function, the domain is all real numbers, because you can plug any $x$-value into the function.

  $\text{Domain} = (-\infty, \infty)$

• Range: Similarly, the range is also all real numbers, because the line extends infinitely in both vertical directions as $x$ increases or decreases.

  $\text{Range} = (-\infty, \infty)$

Would you like further details, such as a graph or more explanation on how the slope works?

Here are 5 related questions to expand on this:

1. What are the key features of a linear function in slope-intercept form?
2. How does changing the slope affect the steepness of the line?
3. How do you find the x-intercept of a linear function?
4. What is the significance of the y-intercept in real-world applications?
5. How can linear functions be used to model real-world scenarios?

Tip: When graphing a line, always start with the y-intercept and use the slope to determine the direction and steepness of the line.