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

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.