Question

If \(3f(x+2)+4f\left(\dfrac{1}{x+2}\right)=4x\), \(x\neq-2\), then \(f(4)\) is:

• A. 7
• B. 52/7
• C. 8
• D. None of the above

Hint:

Replace x with a value that turns 1/(x+2) into the same argument the other term used, so you get a second equation in the same two unknowns.

Answer 1

The Correct Option is D

Instead of solving for the general formula $f(yy)$, we can plug in two specific, well-chosen values of $x$ that both involve $f(44)$, and then solve a small pair of simultaneous equations, since only the value at $4$ is actually needed.

1. First choice, $x=2$: then $x+2=4$ and $\frac{1}{x+2}=\frac{1}{4}$. The relation $3f(x+2)+4f\left(\frac{1}{x+2}\right)=4x$ becomes $$3f(44)+4f\left(\frac{1}{4}\right) = 4(22) = 8 \qquad \text{(II)}$$
2. Second choice, $x=-\frac{7}{4}$: then $x+2 = \frac{1}{4}$ and $\frac{1}{x+2}=4$. The same relation becomes $$3f\left(\frac{1}{4}\right)+4f(44) = 4\left(-\frac{7}{4}\right) = -7 \qquad \text{(II)}$$
3. Eliminate $f\left(\frac{1}{4}\right)$: multiply (II) by $3$ and (II) by $4$: $$9f(44)+12f\left(\frac{1}{4}\right)=24$$ $$16f(44)+12f\left(\frac{1}{4}\right)=-28$$ Subtracting the first from the second gives $7f(44) = -52$, so $f(44)=-\frac{52}{7}$.

This matches the value found by solving for the general function, confirming $f(44)=-\frac{52}{7}$ without ever building the full formula for $f(yy)$.

Let's summarize:

• Choosing $x=2$ and $x=-\frac{7}{4}$ both make the pair $\left(4,\frac{1}{4}\right)$ appear as the two arguments of $f$, letting us treat $f(44)$ and $f\left(\frac{1}{4}\right)$ as two unknowns in two linear equations.
• Solving that pair gives $f(44)=-\frac{52}{7}$, a negative number that is not equal to any of the numeric options 7, 52/7, or 8.

So the correct choice is the option 'None of the above', since $f(44)=-\frac{52}{7}$ is not among the listed positive values.