Choice D is correct. Since $f$ is a linear function, it can be defined by an equation of the form $f (x) = a x + b$ , where $a$ and $b$ are constants. It's given that $f left parenthesis 0 right parenthesis equals 8$ . Substituting $0$ for $x$ and $8$ for $f (x)$ in the equation $f (x) = a x + b$ yields $8 = a (0) + b$ , or $8 equals b$ . Substituting $8$ for $b$ in the equation $f (x) = a x + b$ yields $f (x) = a x + 8$ . It's given that $f left parenthesis 1 right parenthesis equals 12$ . Substituting $1$ for $x$ and $12$ for $f (x)$ in the equation $f (x) = a x + 8$ yields $12 = a (1) + 8$ , or $12 = a + 8$ . Subtracting $8$ from both sides of this equation yields $a equals 4$ . Substituting $4$ for $a$ in the equation $f (x) = a x + 8$ yields $f (x) = 4 x + 8$ . Therefore, an equation that defines $f$ is $f (x) = 4 x + 8$ .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.