Question 733·Medium·Math

A linear function has the form $g(x)=mx+b$, where $m$ is the output change per input step and $b$ is the output at $x=0$. Put each given input and output into that form: $g(2)=-3$ gives $2m+b=-3$, and $g(8)=15$ gives $8m+b=15$. Neither given output is $b$, because neither input is $0$.

Type $[2m+b,8m+b]\sim[-3,15]$ in Desmos. The regression matches the first expression to $-3$ and the second to $15$. Under PARAMETERS, Desmos shows $m=3$ and $b=-9$: the output rises by $3$ for each input step, and its value at zero is $-9$.

Use the numbers Desmos found to type $g(x)=3x-9$. This gives Desmos a fixed function for the next regression. Don't leave $m$ and $b$ in this new line, or Desmos could fit them again. The equation $g(x)=9$ gives the output and asks for its input. Type $g(x_1)\sim9$; $x_1$ tells Desmos to fit an unknown input rather than graph $x$. Under PARAMETERS, it shows $x_1=6$, so the input that gives $g(x)=9$ is $6$. Grid in 6.