SAT Question 02060533: Nonlinear Functions answer and explanation with Desmos solution

Choice A is correct. It's given that the table shows values of $x$ and their corresponding values of $g\left(x\right)$, where $g\left(x\right)=\frac{f\left(x\right)}{x+3}$. It's also given that $f$ is a linear function. It follows that an equation that defines $f$ can be written in the form $f\left(x\right)=mx+b$, where $m$ represents the slope and $b$ represents the y-coordinate of the y-intercept $\left(0,b\right)$ of the graph of $y=f\left(x\right)$ in the xy-plane. The slope of the graph of $y=f\left(x\right)$ can be found using two points, $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$, that are on the graph of $y=f\left(x\right)$, and the formula $m=\frac{y_2-y_1}{x_2-x_1}$. Since the table shows values of $x$ and their corresponding values of $g\left(x\right)$, substituting values of $x$ and $g\left(x\right)$ in the equation $g\left(x\right)=\frac{f\left(x\right)}{x+3}$ can be used to define function $f$. Using the first pair of values from the table, $x=-27$ and $g\left(x\right)=3$, yields $3=\frac{f\left(-27\right)}{-27+3}$, or $3=\frac{f\left(-27\right)}{-24}$. Multiplying each side of this equation by $-24$ yields $-72=f\left(-27\right)$, so the point $\left(-27,-72\right)$ is on the graph of $y=f\left(x\right)$. Using the second pair of values from the table, $x=-9$ and $g\left(x\right)=0$, yields $0=\frac{f\left(-9\right)}{-9+3}$, or $0=\frac{f\left(-9\right)}{-6}$. Multiplying each side of this equation by $-6$ yields $0=f\left(-9\right)$, so the point $\left(-9,0\right)$ is on the graph of $y=f\left(x\right)$. Substituting $\left(-27,-72\right)$ and $\left(-9,0\right)$ for $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$, respectively, in the formula $m=\frac{y_2-y_1}{x_2-x_1}$ yields $m=\frac{0-\left(-72\right)}{-9-\left(-27\right)}$, or $m=4$. Substituting $4$ for $m$ in the equation $f\left(x\right)=mx+b$ yields $f\left(x\right)=4x+b$. Since $0=f\left(-9\right)$, substituting $-9$ for $x$ and $0$ for $f\left(x\right)$ in the equation $f\left(x\right)=4x+b$ yields $0=4\left(-9\right)+b$, or $0=-36+b$. Adding $36$ to both sides of this equation yields $36=b$. It follows that $36$ is the y-coordinate of the y-intercept $\left(0,b\right)$ of the graph of $y=f\left(x\right)$. Therefore, the y-intercept of the graph of $y=f\left(x\right)$ is $\left(0,36\right)$.