The problem asks to find the equation of a linear function $f(x)$ given that:

• $f(-3) = 3$
• $f(1) = -1$

Step 1: Use the slope formula

Since the function is linear, its equation will be in the form:

$f(x) = mx + b$

where $m$ is the slope and $b$ is the y-intercept.

The slope $m$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is calculated by the formula:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Here, the points are $(-3, 3)$ and $(1, -1)$.

Substitute into the formula:

$m = \frac{-1 - 3}{1 - (-3)} = \frac{-4}{4} = -1$

Step 2: Use point-slope form to find the equation

Now that we know the slope $m = -1$, we can use one of the given points to find $b$. Let's use the point $(1, -1)$.

The equation is:

$f(x) = -x + b$

Substitute $x = 1$ and $f(1) = -1$:

$-1 = -(1) + b \implies -1 = -1 + b$

Solving for $b$:

$b = 0$

Final equation:

Thus, the equation for $f(x)$ is:

$f(x) = -x$

Would you like further details on any part of the process?

Here are some related questions:

1. How do you find the slope between two points?
2. What is the significance of the slope in a linear function?
3. Can a linear function have a slope of zero? What would that look like?
4. How do you find the y-intercept from an equation?
5. How can you verify if a point lies on a given line?

Tip: The slope tells you how steep the line is, and whether it increases or decreases as you move from left to right across the graph.