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 $b$.