Once you have the two equations from the points, solve them to find specific values for $m$ and $n$, then rewrite $f(x)$ with those numbers.

To get $g(x)=f\big(\frac{x}{2}+1\big)$, replace every $x$ in your formula for $f(x)$ with $\frac{x}{2}+1$, then simplify the resulting complex fraction.

When you have something like $\dfrac{\frac{x}{2}+7}{\frac{x}{2}-1}$, multiply numerator and denominator by the same number to clear the smaller denominators and match one of the answer choices.

In Desmos, enter f(x) = (x+6)/(x-2) using the $m$ and $n$ values you found. Check that the graph passes through the points (0,-3) and (4,5) to confirm $f(x)$ is correct.

In a new line, enter g(x) = f(x/2 + 1). This uses Desmos’s function composition to create $g(x)$ exactly as defined in the problem.

On separate lines, type the four candidate expressions from choices A–D as functions of $x$ (for example, A(x) = (x+14)/(x-2), B(x) = (x+6)/(x-2), etc.).

Compare the graph of g(x) with the graphs of the four candidate functions. The correct answer is the one whose graph lies exactly on top of g(x) for all visible $x$-values.

We know $f(x)=\dfrac{mx+n}{x-2}$ and that the graph passes through $(0,-3)$, so $f(0)=-3$.

Substitute $x=0$ into $f(x)$:

Solve for $n$:

Now use the second point $(4,5)$, so $f(4)=5$.

Substitute $x=4$ into $f(x)$, using $n=6$:

Solve for $m$:

With $m=1$ and $n=6$, substitute back into the definition of $f$:

Now we will use this $f(x)$ to build $g(x)$ by composition.

By definition,

Replace $x$ in $f(x)=\dfrac{x+6}{x-2}$ with $\dfrac{x}{2}+1$:

To clear the fractions in numerator and denominator, multiply both by $2$:

So the correct expression is $g(x)=\dfrac{x+14}{x-2}$, which corresponds to choice A.