f ( x ) = 4 x + b For the linear function f , b

Choice A is correct. For the linear function $f$, it’s given that $f\left(7\right)=28$. Substituting $7$ for $x$ and $28$ for $f\left(x\right)$ in the given function yields $28=4\left(7\right)+b$, or $28=28+b$. Subtracting $28$ from each side of this equation yields $0=b$. Therefore, the value of $b$ is $0$.

Choice B is incorrect. Substituting $1$ for $b$ in the given function yields $f\left(x\right)=4x+1$. For this function, when the value of $x$ is $7$, the value of $f\left(x\right)$ is $29$, not $28$.

Choice C is incorrect. Substituting $4$ for $b$ in the given function yields $f\left(x\right)=4x+4$. For this function, when the value of $x$ is $7$, the value of $f\left(x\right)$ is $32$, not $28$.

Choice D is incorrect. Substituting $7$ for $b$ in the given function yields $f\left(x\right)=4x+7$. For this function, when the value of $x$ is $7$, the value of $f\left(x\right)$ is $35$, not $28$.