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

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

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

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