Question 199·200 Super-Hard SAT Math Questions·Advanced Math

Type $f(x)=2^{4x-1}\cdot8^{1-2x}$ into Desmos. A function is a rule that gives an output for an input. Keep both exponents as written; defining $f$ lets you use it at different inputs without retyping the rule.

Type $f(x_1+1)\sim16f(3x_1)$. The whole $x_1+1$ goes into the first $f$, and the whole $3x_1$ goes into the second. Use $x_1$ instead of the graphing variable $x$; the regression symbol $\sim$ tells Desmos to find its value. Under PARAMETERS, Desmos shows $x_1=1.5$. Since $1.5=\frac32$, the value of $x$ that satisfies the given equation is $\frac32$. Choice C.

The base is the number being raised to a power. Since $8=2^3$, write both factors with base $2$. Let $t$ stand for any input: $f(t)=2^{4t-1}\cdot(2^3)^{1-2t}$.

A power of a power multiplies its exponents. The $3$ multiplies the whole $1-2t$: $f(t)=2^{4t-1}\cdot2^{3(1-2t)}$.

Multiplying powers with the same base adds their exponents. Add them: $f(t)=2^{4t-1+3(1-2t)}$. Distribute the $3$: $f(t)=2^{4t-1+3-6t}$. Combine like terms: $f(t)=2^{2-2t}$.

In $f(x+1)$, $x+1$ is the entire input, so replace $t$ with $x+1$: $f(x+1)=2^{2-2(x+1)}$. Distribute the $-2$: $f(x+1)=2^{-2x}$.

For $f(3x)$, replace $t$ with $3x$, not with $x$ and then cube the output: $f(3x)=2^{2-2(3x)}$. Multiply: $f(3x)=2^{2-6x}$. The constant $2$ in the exponent hasn't been tripled.

The $16$ multiplies $f(3x)$, and $16=2^4$. Substitute the two outputs into the given equation: $2^{-2x}=2^4\cdot2^{2-6x}$. Add the exponents on the right: $2^{-2x}=2^{6-6x}$.

Equal powers of base $2$ have equal exponents, so set them equal: $-2x=6-6x$. Add $6x$ to both sides: $4x=6$. Divide both sides by $4$: $x=\frac64=\frac32$. So $\frac32$ is the input that makes $f(x+1)$ equal $16f(3x)$. Choice C.