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).