To find the linear function $f(x)$ based on the given values $f(-4) = -5$ and $f(1) = 4$, we can use the point-slope form of a linear equation. A linear function can be written as:

$f(x) = mx + b$

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

Step 1: Calculate the slope

We know two points on the line: $(-4, -5)$ and $(1, 4)$. The formula for the slope $m$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

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

Substitute the given points:

$m = \frac{4 - (-5)}{1 - (-4)} = \frac{4 + 5}{1 + 4} = \frac{9}{5}$

So, the slope is $m = \frac{9}{5}$.

Step 2: Find the y-intercept

We can now use the slope-intercept form $f(x) = mx + b$ and substitute one of the points to solve for $b$. Using the point $(1, 4)$:

$4 = \frac{9}{5}(1) + b$

Solve for $b$:

$4 = \frac{9}{5} + b$

Multiply both sides by 5 to eliminate the fraction:

$20 = 9 + 5b$

Subtract 9 from both sides:

$11 = 5b$

Now, solve for $b$:

$b = \frac{11}{5}$

Step 3: Write the equation

Now that we have the slope $m = \frac{9}{5}$ and the y-intercept $b = \frac{11}{5}$, the linear function is:

$f(x) = \frac{9}{5}x + \frac{11}{5}$

Final Answer:

$f(x) = \frac{9}{5}x + \frac{11}{5}$

Would you like further details on any of the steps?

Here are some related questions to further your understanding:

1. How do you convert a linear equation from point-slope form to slope-intercept form?
2. What is the significance of the slope in a linear function?
3. How can you graph a linear equation using only two points?
4. How do you determine the x-intercept of a linear function?
5. How would the function change if we modified the slope or the y-intercept?

Tip: When given two points, always calculate the slope first as it's the key to forming the linear equation.