To solve the linear function $f(x)$ given the points $f(-3) = 4$ and $f(2) = 7$, we can start by using the slope-intercept form of a linear equation:

$f(x) = mx + b$

Step 1: Find the slope $m$

The slope $m$ of a linear function is calculated using the formula:

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

where $(x_1, y_1) = (-3, 4)$ and $(x_2, y_2) = (2, 7)$.

Substituting the values:

$m = \frac{7 - 4}{2 - (-3)} = \frac{3}{5} = 0.6$

Step 2: Use point-slope form to find $b$

Now that we have the slope $m = 0.6$, we can substitute it into the equation $f(x) = mx + b$. Using one of the points, for example $(-3, 4)$, we solve for $b$.

$4 = 0.6(-3) + b$ $4 = -1.8 + b$ $b = 4 + 1.8 = 5.8$

Step 3: Write the equation of the linear function

Now that we have both the slope $m = 0.6$ and the intercept $b = 5.8$, the equation of the linear function is:

$f(x) = 0.6x + 5.8$

This is the linear function for $f(x)$.

Do you have any further questions or need more details? Here are some related questions to explore:

1. How do you determine the slope between two points?
2. What is the significance of the slope in a linear function?
3. How do you find the y-intercept of a linear function?
4. Can a linear function have a slope of zero? What does that mean graphically?
5. How would the equation change if one of the points was different?

Tip: Whenever solving for the equation of a linear function, always calculate the slope first as it helps set up the rest of the function structure.