Find f(0) and then find the equation of the given linear function.
X = 1, 2, 3, 4 and  f(x) = 6, 9, 12, 15. Solve for f(0) and f(x)

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.

Learn how to find f(0) and the equation of the linear function f(x) = 3x + 3 with step-by-step solutions. This problem involves calculating the slope, determining the y-intercept, and applying the linear function formula. Suitable for students in Grades 7-9.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation