The question from your image states:

Given:

• $f(x)$ is a linear function.
• $f(-3) = 4$
• $f(0) = 3$

Find: An equation for $f(x)$.

Solution:

Since $f(x)$ is a linear function, it can be written in the form:

$f(x) = mx + b$

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

Step 1: Use Given Points to Set Up Equations

We have two points:

1. $(-3, 4)$ from $f(-3) = 4$
2. $(0, 3)$ from $f(0) = 3$

Substitute these points into the linear equation:

1. For $f(-3) = 4$:
   $4 = m(-3) + b$

2. For $f(0) = 3$:
   $3 = m(0) + b \implies b = 3$

Step 2: Solve for Slope $m$

Substitute $b = 3$ into the first equation:

$4 = m(-3) + 3$

Simplify to find $m$:

$4 - 3 = -3m \implies 1 = -3m \implies m = -\frac{1}{3}$

Step 3: Write the Equation

Now substitute $m = -\frac{1}{3}$ and $b = 3$ back into the linear equation form:

$f(x) = -\frac{1}{3}x + 3$

Final Answer:

The equation for $f(x)$ is:

$\boxed{f(x) = -\frac{1}{3}x + 3}$

Would you like more details or have any other questions?

Follow-up Questions:

1. How do you find the slope of a linear function given two points?
2. What is the significance of the y-intercept in a linear equation?
3. How do you graph a linear function on the coordinate plane?
4. How can you determine if a function is linear or non-linear?
5. How does the slope affect the steepness and direction of a line?

Tip:

For linear functions, knowing two points is sufficient to determine the equation of the line.