To solve for $f(0)$ and the equation of the linear function $f(x)$, follow these steps:

Step 1: Identify the pattern in the function values.

Given:

• $f(1) = 6$
• $f(2) = 9$
• $f(3) = 12$
• $f(4) = 15$

Step 2: Calculate the slope ($m$) of the linear function.

The slope $m$ of a linear function is given by the formula: $m = \frac{\Delta f(x)}{\Delta x}$ Using any two points, say $(1, 6)$ and $(2, 9)$: $m = \frac{9 - 6}{2 - 1} = \frac{3}{1} = 3$

Step 3: Determine the equation of the linear function $f(x)$.

The general form of a linear function is: $f(x) = mx + b$ Substitute $m = 3$ into the equation: $f(x) = 3x + b$

Step 4: Solve for the y-intercept ($b$).

Use one of the given points, for example, $(1, 6)$: $6 = 3(1) + b$ $6 = 3 + b \implies b = 3$ Thus, the equation of the function is: $f(x) = 3x + 3$

Step 5: Find $f(0)$.

Substitute $x = 0$ into the equation $f(x) = 3x + 3$: $f(0) = 3(0) + 3 = 3$

Final Results:

• The equation of the linear function is $f(x) = 3x + 3$.
• $f(0) = 3$.

Would you like more details or have any questions?

Here are some related questions you might consider:

1. How do we generalize finding the slope for any set of points?
2. What happens if the points do not form a linear relationship?
3. How can we verify that the function $f(x) = 3x + 3$ fits all given points?
4. How do we find the x-intercept of this linear function?
5. How does the slope $m$ affect the steepness of the line?

Tip: Always verify the function with multiple points to ensure the accuracy of the derived equation.