SAT Question b0543157: Nonlinear Functions answer and explanation with Desmos solution

Choice A is correct. It's given that in the xy-plane, the graph of $y=g(x)$ passes through the point $(0,22)$. It follows that $g(0)=22$. Substituting $0$ for $x$ and $22$ for $g(x)$ in the given equation yields $22=\frac{0^2-0-a}{0^3-0-b}$, which is equivalent to $22=\frac{-a}{-b}$, or $22=\frac{a}{b}$. Multiplying each side of this equation by $b$ yields $22b=a$. Substituting $22b$ for $a$ in the equation defining the function $g$ yields $g\left(x\right)=\frac{x^2-x-22b}{x^3-x-b}$. It's given that $g(-22)=0$. Substituting $-22$ for $x$ and $0$ for $g(x)$ in the equation $g\left(x\right)=\frac{x^2-x-22b}{x^3-x-b}$ yields $0=\frac{(-22{)}^2-(-22)-22b}{(-22{)}^3-(-22)-b}$. Since a rational expression equals $0$ if and only if its numerator is $0$, it follows that $(-22{)}^2-\left(-22\right)-22b=0$, or $22^2+22-22b=0$. Dividing each side of this equation by $22$ yields $22+1-b=0$, or $23-b=0$. Adding $b$ to each side of this equation yields $23=b$, so the value of $b$ is $23$.