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

- \( f(-2) = -5 \)
- \( f(3) = 3 \)

### Step-by-step solution:

1. A linear function has the general form:
   \[
   f(x) = mx + b
   \]
   where \( m \) is the slope, and \( b \) is the y-intercept.

2. To find the slope \( m \), we use the slope formula:
   \[
   m = \frac{y_2 - y_1}{x_2 - x_1}
   \]
   Using the points \( (-2, -5) \) and \( (3, 3) \):
   \[
   m = \frac{3 - (-5)}{3 - (-2)} = \frac{3 + 5}{3 + 2} = \frac{8}{5}
   \]

3. Now that we have \( m = \frac{8}{5} \), substitute this into the linear function:
   \[
   f(x) = \frac{8}{5}x + b
   \]

4. To find \( b \), substitute one of the given points (e.g., \( (-2, -5) \)) into the equation:
   \[
   -5 = \frac{8}{5}(-2) + b
   \]
   Simplifying:
   \[
   -5 = \frac{8}{5} \times (-2) + b = \frac{-16}{5} + b
   \]
   Multiply both sides by 5 to eliminate the fraction:
   \[
   -25 = -16 + 5b
   \]
   Solve for \( b \):
   \[
   -25 + 16 = 5b \quad \Rightarrow \quad -9 = 5b \quad \Rightarrow \quad b = -\frac{9}{5}
   \]

5. Therefore, the equation of the linear function is:
   \[
   f(x) = \frac{8}{5}x - \frac{9}{5}
   \]

Would you like further clarification or more details?

### Related Questions:
1. How do you verify if a function is linear?
2. Can you explain how to graph a linear function from its equation?
3. What happens to the slope if the points are reversed?
4. How would the equation change if \( f(x) \) were a quadratic function?
5. What does the slope represent in real-world problems?

### Tip:
When solving for the y-intercept in a linear function, substituting any given point into the equation ensures you accurately determine the correct value of \( b \).

If f(x) is a linear function, f(-2) = -5, and f(3) = 3, find an equation for f(x).

Learn how to find the equation of a linear function f(x) given two points (-2, -5) and (3, 3). This problem involves calculating the slope and determining the y-intercept to form the linear equation in slope-intercept form. Suitable for students in Grades 8-10.

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