Đáp án: $f\left( 4 \right) =  - 14$

Giải thích các bước giải:

$\begin{array}{l}
f\left( x \right) = a.x + b\\
 \Leftrightarrow \left\{ \begin{array}{l}
f\left( 2 \right) = 2a + b\\
f\left( 3 \right) = 3a + b\\
f\left( 1 \right) = a + b\\
f\left( { - 1} \right) =  - a + b
\end{array} \right.\\
Khi:\left\{ \begin{array}{l}
2f\left( 2 \right) - f\left( 3 \right) = 4\\
5f\left( 1 \right) - f\left( { - 1} \right) = 4\\
2f\left( 2 \right) - f\left( 3 \right) = 5f\left( 1 \right) - f\left( { - 1} \right)
\end{array} \right.\\
 \Leftrightarrow \left\{ \begin{array}{l}
2.\left( {2a + b} \right) - \left( {3a + b} \right) = 4\\
5.\left( {a + b} \right) - \left( { - a + b} \right) = 4\\
2.\left( {2a + b} \right) - \left( {3a + b} \right) = 5.\left( {a + b} \right) - \left( { - a + b} \right)
\end{array} \right.\\
 \Leftrightarrow \left\{ \begin{array}{l}
4a + 2b - 3a - b = 4\\
5a + 5b + a - b = 4\\
4a + 2b - 3a - b = 5a + 5b + a - b
\end{array} \right.\\
 \Leftrightarrow \left\{ \begin{array}{l}
a + b = 4\\
6a + 4b = 4\\
a + b = 6a + 4b
\end{array} \right.\\
 \Leftrightarrow \left\{ \begin{array}{l}
2a + 2b = 8\\
3a + 2b = 2\\
5a =  - 3b
\end{array} \right.\\
 \Leftrightarrow \left\{ \begin{array}{l}
a =  - 6\\
b = 4 - a = 10\\
5.\left( { - 6} \right) =  - 3.10\left( {tm} \right)
\end{array} \right.\\
 \Leftrightarrow a =  - 6;b = 10\\
f\left( 4 \right) = a.4 + b =  - 6.4 + 10 =  - 14
\end{array}$