Functional‑Equation “Plug x & 1 − x” Technique

Fast Solution

1. Write the given equation for $x$

   $$ f(x)+2f(1-x)=3x.\tag{1} $$

2. Write the same equation for $1-x$

   $$ f(1-x)+2f(x)=3(1-x)=3-3x.\tag{2} $$

3. Solve the 2 × 2 linear system in $f(x)$ and $f(1-x)$.

   Multiply (1) by 2 and subtract (2):

   $$ \begin{aligned} 2f(x)+4f(1-x) &= 6x \[2pt] \bigl(2f(x)+4f(1-x)\bigr) - \bigl(f(1-x)+2f(x)\bigr) &= 6x - (3-3x) \[2pt] 3f(1-x) &= 9x - 3 \[2pt] f(1-x) &= 3x - 1. \end{aligned} $$

   Substitute $f(1-x)=3x-1$ back into (1):

   $$ f(x) + 2(3x-1) = 3x ;\Longrightarrow; f(x) = -3x + 2. $$

4. Evaluate at $x=\dfrac34$.

   $$ f!\left(\frac34\right) = -3!\left(\frac34\right)+2 = -\frac94 + 2 = -\frac14. $$