Question-27

For \(\displaystyle x\in \mathbb{R}\), the function \(\displaystyle f( x)\) satisfies the following identity:

\[ \begin{equation*} 2f( x) +f( 1-x) =x^{2} \end{equation*} \]

Find \(\displaystyle f( 4)\).

Hint

Substitute \(x \rightarrow 1 - x\)

Answer

\(\cfrac{23}{3}\)

Solution

Substituting \(\displaystyle x\rightarrow 1-x\), we get:

\[ \begin{equation*} 2f( 1-x) +f( x) =( 1-x)^{2} \end{equation*} \]

Using this in conjunction with the original identity, we have:

\[ \begin{equation*} 2f( x) -\frac{f( x)}{2} =x^{2} -\frac{( 1-x)^{2}}{2} \end{equation*} \]

From this, we get:

\[ \begin{equation*} f( x) =\frac{1}{3}\left[ x^{2} +2x-1\right] \end{equation*} \]

Plugging in \(\displaystyle x=4\), we get \(\displaystyle f( 4) =\frac{23}{3}\).