For the linear function g , the graph of y = g ( x ) in

Choice C is correct. An equation defining a linear function can be written in the form $g\left(x\right)=mx+b$, where $m$ is the slope and $\left(0,b\right)$ is the y-intercept of the graph of $y=g\left(x\right)$ in the xy-plane. It’s given that the graph of $y=g\left(x\right)$ has a slope of $2$. Therefore, $m=2$. It’s also given that the graph of $y=g\left(x\right)$ passes through the point $\left(1,14\right)$. It follows that when $x=1$, $g\left(x\right)=14$. Substituting $1$ for $x$, $14$ for $g\left(x\right)$, and $2$ for $m$ in the equation $g\left(x\right)=mx+b$ yields $14=2\left(1\right)+b$, or $14=2+b$. Subtracting $2$ from each side of this equation yields $12=b$. Therefore, $b=12$. Substituting $2$ for $m$ and $12$ for $b$ in the equation $g\left(x\right)=mx+b$ yields $g\left(x\right)=2x+12$. Therefore, the equation that defines $g$ is $g\left(x\right)=2x+12$.

Choice A is incorrect. For this function, the graph of $y=g\left(x\right)$ in the xy-plane passes through the point $\left(1,2\right)$, not $\left(1,14\right)$.

Choice B is incorrect. For this function, the graph of $y=g\left(x\right)$ in the xy-plane passes through the point $\left(1,4\right)$, not $\left(1,14\right)$.

Choice D is incorrect. For this function, the graph of $y=g\left(x\right)$ in the xy-plane passes through the point $\left(1,16\right)$, not $\left(1,14\right)$.