Question 250·Hard·Math

Type $f(x)=2^{x-1}+2^{1-x}$ in Desmos and click its lowest point. Desmos shows $(1,2)$: $1$ is the input, and $2$ is the minimum, or smallest output. At input $1$, both exponents are $0$, which helps you recognize the point.

Type $g(x)=5-3f(2x+4)$ below $f$. Desmos graphs a peak for $g$. The negative factor $-3$ flips the outputs: the smallest $f$ output gives the largest $g$ output. Adding $5$ changes their height, not which input produces the peak.

The lowest point of $f$ uses input $1$. Since $g$ feeds the whole expression $2x+4$ into $f$, set that expression equal to $1$: $2x+4=1$. Match the inside expression to the original function’s key input; using $x=1$ would ignore the changed input.

In Desmos, replace $x$ with $x_1$ and $=$ with $\sim$: type $2x_1+4\sim1$. The $\sim$ asks Desmos to find the value of $x_1$ that makes the sides equal. Under PARAMETERS, it shows $x_1=-1.5$. Since $-1.5=-\frac{3}{2}$, $g$ reaches its maximum at $x=-\frac{3}{2}$. Choice C.