To find the equation for the linear function $f(x)$, we need to determine the slope $m$ and the y-intercept $b$ of the line. The general form of a linear equation is:

$f(x) = mx + b$

Step 1: Find the slope $m$

The slope $m$ is calculated as the change in $f(x)$ over the change in $x$. We are given two points: $(-2, -3) \quad \text{and} \quad (4, -5)$ Using the slope formula: $m = \frac{f(4) - f(-2)}{4 - (-2)} = \frac{-5 - (-3)}{4 - (-2)} = \frac{-5 + 3}{4 + 2} = \frac{-2}{6} = -\frac{1}{3}$

Step 2: Find the y-intercept $b$

Now that we know $m = -\frac{1}{3}$, we can use one of the given points to solve for $b$. Let's use the point $(-2, -3)$.

Substitute $x = -2$, $f(x) = -3$, and $m = -\frac{1}{3}$ into the equation: $-3 = -\frac{1}{3}(-2) + b$ Simplify: $-3 = \frac{2}{3} + b$ Subtract $\frac{2}{3}$ from both sides: $b = -3 - \frac{2}{3} = \frac{-9}{3} - \frac{2}{3} = \frac{-11}{3}$

Step 3: Write the equation

Now that we know $m = -\frac{1}{3}$ and $b = \frac{-11}{3}$, the equation for $f(x)$ is: $f(x) = -\frac{1}{3}x - \frac{11}{3}$

Would you like more details or have any questions?

Here are 5 related questions:

1. How do you verify the correctness of the linear equation using given points?
2. What are the steps for finding the slope between two points?
3. How do you graph a linear function given the equation?
4. How would the equation change if the slope were positive?
5. What is the significance of the y-intercept in a linear equation?

Tip: Always double-check your slope calculation by ensuring you subtract the values of $f(x)$ and $x$ in the correct order!