Once you have $f\!\left(\dfrac{x}{2} + 1\right)$, multiply it by $\dfrac{1}{60}$ to get a simplified expression for $g(x)$. Focus on simplifying the numerical coefficient first.

Your expression for $g(x)$ will likely involve $5^{x-1}$. Use exponent rules to rewrite $5^{x-1}$ in terms of $5^x$, then combine all constants so it looks like $k \cdot 5^x$, like the answer choices.

In Desmos, type f(x) = 12*5^(2x-3). This stores the given function f so you can reuse it.

On a new line, type g(x) = (1/60)*f(x/2 + 1). This creates the function g exactly as defined in the problem.

On additional lines, enter four functions corresponding to the right-hand sides of the answer choices, for example: h1(x) = (12/5)*5^x, h2(x) = (1/25)*5^x, h3(x) = (5/12)*5^x, and h4(x) = (5/60)*25^x. These represent each possible expression for g(x).

Look at the graphs of g(x) and h1, h2, h3, h4. The correct answer is the one whose graph lies exactly on top of the graph of g(x) for all x-values (they should be indistinguishable). Read off which expression that is from your Desmos definitions.

We are told that

and that

First find an explicit expression for $f\!\left(\frac{x}{2} + 1\right)$.

Replace $x$ in $f(x)$ with $\dfrac{x}{2} + 1$:

So $f\!\left(\frac{x}{2} + 1\right) = 12 \cdot 5^{x-1}$.

Now plug this into the definition of $g$:

Simplify the constant factor:

So we have

This is very close to the form $k \cdot b^x$, but the exponent is $x - 1$ instead of $x$, so we need to rewrite it.

Use exponent rules to rewrite $5^{x-1}$ in terms of $5^x$:

Substitute this into the expression for $g(x)$:

Now $g$ is in the form $g(x) = k \cdot b^x$ with $k = \dfrac{1}{25}$ and $b = 5$, which matches choice B: $g(x) = \dfrac{1}{25} (5)^x$.