Question 23·SAT Desmos Regression Practice·Algebra

The equation says an input increase of $3$ raises the output by $5$. For a linear function, the slope is output change divided by input change, so $m=\frac{5}{3}$. This rate works between any two inputs, not only inputs exactly $3$ apart.

Let $t=f(2)$. In a composition, the inner output becomes the outer input, so $f(f(2))=34$ means $f(t)=34$. From input $2$ to input $t$, the input changes by $t-2$. The output starts at $f(2)=t$, so the slope gives $34=t+\frac{5}{3}(t-2)$. The same $t$ names the first output and the next input.

Type $t+\frac{5}{3}(t-2)\sim34$. The regression symbol $\sim$ asks Desmos to find the unknown rather than graph the equation. Under PARAMETERS, Desmos shows $t=14$, so $f(2)=14$.

From input $2$ to input $11$ takes three jumps of $3$, so the output rises by $5$ three times. Type $14+3(5)$ on the next Desmos line; it prints $29$. The given $34$ is $f(14)$, not $f(11)$. So $f(11)=29$. Choice B.