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To determine which equation defines the linear function \( f \), given the conditions \( f(0) = 8 \) and \( f(1) = 12 \), we need to find the equation of the line in the form \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

1. Use the given condition \( f(0) = 8 \):
\[ f(0) = 8 \Rightarrow b = 8 \]
So, the equation so far is:
\[ f(x) = mx + 8 \]

2. Use the second condition \( f(1) = 12 \):
\[ f(1) = m(1) + 8 = 12 \]
\[ m + 8 = 12 \]
\[ m = 4 \]

Therefore, the equation of the linear function is:
\[ f(x) = 4x + 8 \]

So, the correct answer is:
\[ \boxed{D} \]

Do you want more details or have any questions?

Here are some related questions for further practice:

1. If \( g(0) = 5 \) and \( g(2) = 11 \), what is the equation of the linear function \( g(x) \)?
2. For the function \( h(x) \) such that \( h(0) = -3 \) and \( h(1) = 2 \), find the equation of \( h(x) \).
3. Given \( j(0) = 7 \) and \( j(2) = 15 \), what is the slope of the function \( j(x) \)?
4. If \( k(0) = 1 \) and \( k(3) = 10 \), determine \( k(x) \).
5. Find the equation of the line that passes through the points \((0, 4)\) and \((2, 10)\).
6. Given \( m(0) = 6 \) and \( m(5) = 21 \), what is the equation of \( m(x) \)?
7. For the function \( p(x) \) such that \( p(0) = -2 \) and \( p(3) = 7 \), find the slope of \( p(x) \).
8. If \( q(0) = 0 \) and \( q(4) = 16 \), determine the equation of \( q(x) \).

**Tip:** When finding the equation of a line given two points, first find the slope (m) using \( m = \frac{y_2 - y_1}{x_2 - x_1} \) and then use one of the points to solve for the y-intercept (b).

Learn how to solve the linear function equation f(x) = 4x + 8 with detailed steps and explanations. This problem covers linear functions and equations of lines, including finding the slope and y-intercept. Suitable for students in Grades 7-9.

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