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

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

Step 1: Graphing the Function

This is a linear function in the slope-intercept form $f(x) = mx + b$, where:

• $m = -5$ is the slope, indicating the steepness and direction of the line.
• $b = 3$ is the y-intercept, where the line crosses the y-axis.

1. Plot the y-intercept: The point $(0, 3)$ is on the graph.
2. Use the slope: Since the slope is $-5$, for every 1 unit increase in $x$, $f(x)$ decreases by 5 units. So from $(0, 3)$, move 1 unit to the right and 5 units down to plot another point $(1, -2)$.
3. Draw the line: Connect the points to extend the line.

Step 2: Domain and Range

• Domain: The domain of a linear function is all real numbers because the function is defined for any value of $x$. $\text{Domain} = (-\infty, \infty)$
• Range: Similarly, the range of a linear function is all real numbers because as $x$ approaches infinity or negative infinity, $f(x)$ can take any value. $\text{Range} = (-\infty, \infty)$

Would you like me to plot the graph for you or explain further details?

Related Questions:

1. How do you calculate the slope from two points?
2. How can the slope help predict changes in the function?
3. What happens if the slope is zero or positive?
4. How do you find intercepts for other types of functions?
5. Can you graph this function without knowing the slope?

Tip:

When dealing with linear functions, always identify the slope and intercept first, as they guide the graphing process efficiently.