The text in the image appears to ask for the equation of a linear function $f(x)$, given certain values of the function for specific $x$-values. Here's the problem extracted and analyzed:

Problem:

A linear function $f$ satisfies the following conditions:

• $f(x) = -6$ when $x = 3$
• $f(x) = 8$ when $x = -4$

We need to determine the equation for the function $f(x)$.

Solution:

The general form of a linear function is: $f(x) = mx + c$ where $m$ is the slope and $c$ is the y-intercept.

1. Find the slope $m$:
   The slope $m$ can be found using the formula: $m = \frac{f(x_2) - f(x_1)}{x_2 - x_1}$ Using the given points $(3, -6)$ and $(-4, 8)$: $m = \frac{8 - (-6)}{-4 - 3} = \frac{8 + 6}{-7} = \frac{14}{-7} = -2$

2. Write the partial equation:
   Substitute $m = -2$ into the general equation: $f(x) = -2x + c$

3. Find the y-intercept $c$:
   Use one of the given points to solve for $c$. Using $(3, -6)$: $-6 = -2(3) + c$ $-6 = -6 + c$ $c = 0$

4. Final equation:
   The equation for the linear function is: $f(x) = -2x$

Verification:

• For $x = 3$: $f(3) = -2(3) = -6$ ✅
• For $x = -4$: $f(-4) = -2(-4) = 8$ ✅

Thus, the equation $f(x) = -2x$ satisfies both conditions.

Would you like additional details or explanations on any step? 😊

Related Questions:

1. What is the slope-intercept form of a linear equation?
2. How can you derive the slope formula for two points?
3. Can a linear function have more than one solution for $f(x)$ for the same $x$?
4. How would the function change if $c \neq 0$?
5. What is the significance of the y-intercept in a linear equation?

Tip:

Always verify your final function with the given points to ensure accuracy!