Question

If \(f(x)+3f(1-x)=x+4\), then \(f(x)=\)

• A. \(\frac{19-4x}{8}\)
• B. \(\frac{11-4x}{8}\)
• C. \(\frac{11-2x}{8}\)
• D. \(\frac{11-2x}{9}\)
• E. \(\frac{11-4}{9}\)

Hint:

In functional equations involving \(f(x)\) and \(f(1-x)\), always substitute \(x \to 1-x\) to create a system and solve simultaneously.

Answer 1

The Correct Option is B

Step 1: Write the given functional equation.
\[ f(x)+3f(1-x)=x+4 \]

Step 2: Replace \(x\) by \(1-x\).
\[ f(1-x)+3f(x)=1-x+4 \] \[ f(1-x)+3f(x)=5-x \]

Step 3: Form a system of equations.
We now have:
\[ (1)\quad f(x)+3f(1-x)=x+4 \] \[ (2)\quad 3f(x)+f(1-x)=5-x \]

Step 4: Eliminate one variable.
Multiply equation (1) by \(3\):
\[ 3f(x)+9f(1-x)=3x+12 \]

Step 5: Subtract equation (2).
\[ (3f(x)+9f(1-x))-(3f(x)+f(1-x))=(3x+12)-(5-x) \] \[ 8f(1-x)=4x+7 \] \[ f(1-x)=\frac{4x+7}{8} \]

Step 6: Substitute back to find \(f(x)\).
Substitute into equation (1):
\[ f(x)+3\cdot \frac{4x+7}{8}=x+4 \] \[ f(x)=x+4-\frac{12x+21}{8} \] \[ f(x)=\frac{8x+32-12x-21}{8} \] \[ f(x)=\frac{11-4x}{8} \]

Step 7: Final answer.
Thus,
\[ \boxed{f(x)=\frac{11-4x}{8}} \] which matches option \((2)\).