To find the equation of the linear function $f(x)$, we can use the general form of a linear function:

$f(x) = mx + b$

where $m$ is the slope and $b$ is the y-intercept. Given the points $(-4, 3)$ and $(5, -5)$, we can determine the slope $m$ using the formula:

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

Substitute the given points into the formula:

$m = \frac{-5 - 3}{5 - (-4)} = \frac{-5 - 3}{5 + 4} = \frac{-8}{9} = -\frac{8}{9}$

Now that we have the slope $m = -\frac{8}{9}$, we can use one of the points to find the y-intercept $b$. Let's use the point $(-4, 3)$:

$f(x) = -\frac{8}{9}x + b$

Substitute $x = -4$ and $f(-4) = 3$:

$3 = -\frac{8}{9}(-4) + b$

$3 = \frac{32}{9} + b$

To isolate $b$, subtract $\frac{32}{9}$ from both sides:

$3 - \frac{32}{9} = b$

Convert 3 to a fraction with a denominator of 9:

$3 = \frac{27}{9}$

So,

$\frac{27}{9} - \frac{32}{9} = b$

$b = \frac{27 - 32}{9}$

$b = -\frac{5}{9}$

Therefore, the equation of the linear function is:

$f(x) = -\frac{8}{9}x - \frac{5}{9}$

Would you like any further details or have any questions about this solution?

Here are 5 questions you might ask next:

1. How do I find the slope of a line from two points?
2. Can you explain how to convert a decimal to a fraction?
3. What is the significance of the y-intercept in a linear function?
4. How do you graph a linear function given its equation?
5. Can you show me how to solve a system of linear equations?

Tip: When solving for the y-intercept $b$ in a linear equation, always substitute one of the given points into the equation after determining the slope to solve for $b$ accurately.