Question 666·Medium·Math

$f(x)$ is the output produced by input $x$. Since $f(x)=61$ gives the output, replace $f(x)$ with its rule:

Type $f(x)=7\cdot2^{x-3}+5$ to define the function, then type $f(x_1)\sim61$. Desmos uses $x_1$ as the number to find rather than an input to graph; $\sim$ asks it to match the two sides. Under PARAMETERS, Desmos shows $x_1=6$. So the input that makes the output $61$ is $x=6$. Choice C.

Starting from $7\cdot2^{x-3}+5=61$, subtract $5$ from both sides to remove the outside addition:

Divide both sides by $7$ so the power of $2$ stands alone:

The base is the number being raised to a power. Since $8=2^3$, rewrite the right side with base $2$:

Equal powers of the same base greater than $1$ have equal exponents. The bases now match, so set the exponents equal:

Add $3$ to both sides to undo the subtraction in the exponent:

So the input that produces an output of $61$ is $6$. Choice C.