The linear function g is defined by g x = b - 15 x , where

Choice C is correct. It’s given that $g\left(c+7\right)=\frac{c}{4}$. Therefore, for the given linear function $g$, when $x=c+7$, $g\left(x\right)=\frac{c}{4}$. Substituting $c+7$ for $x$ and $\frac{c}{4}$ for $g\left(x\right)$ in $g\left(x\right)=b-15x$ yields $\frac{c}{4}=b-15\left(c+7\right)$. Applying the distributive property to the right-hand side of this equation yields $\frac{c}{4}=b-15c-105$. Adding $15c$ to both sides of this equation yields $\frac{c}{4}+15c=b-105$. Adding $105$ to both sides of this equation yields $\frac{c}{4}+15c+105=b$, or $\frac{61c}{4}+105=b$. Therefore, the expression that represents the value of $b$ is $\frac{61c}{4}+105$.

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 D is incorrect and may result from conceptual or calculation errors.