Discussion
If \(f(x) + 3f(1-x) = x + 4\), then \(f(x)\)=
Answer: B
\(f(x)\) is \(\frac{11 - 4x}{8}\).
Replace x by \(1 - x\) to get a second equation, then solve the pair:
\[ \begin{aligned} f(x) + 3f(1 - x) &= x + 4 \\ f(1 - x) + 3f(x) &= 5 - x \end{aligned} \]
Multiply the second by 3 and subtract the first: \(9f(x) - f(x) = 15 - 3x - x - 4\), so \(8f(x) = 11 - 4x\).
Check at x = 0: f(0) = 11/8 and f(1) = 7/8, and \(\frac{11}{8} + 3 \times \frac{7}{8} = 4\), as required. \(\frac{19 - 4x}{8}\) and the options over 9 fail this check; for example \(\frac{19}{8} + 3 \times \frac{15}{8}\) is far from 4.
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