Question

Let $f:R \to R$ be defined by $f\left( x \right) = \left\{ \begin{gathered}
  2x{\text{ }}x > 3 \\
  {x^2}{\text{ }}1 < x \leqslant 3 \\
  3x{\text{ }}x \leqslant 1 \\
\end{gathered} \right\}$. Then what is the value of $f\left( { - 1} \right) + f\left( 2 \right) + f\left( 4 \right)$?
$
  {\text{A}}{\text{. 9}} \\
  {\text{B}}{\text{. 14}} \\
  {\text{C}}{\text{. 5}} \\
  {\text{D}}{\text{. 10}} \\
 $

Answer 1

Hint- Here, we will proceed by finding the values of the function for some specific values of x (in this case, we will find the values of the function corresponding to x = -1, x = 2 and x = 4) according to the definition of the function given.

Complete step-by-step answer:
The given function is defined as $f\left( x \right) = \left\{ \begin{gathered}
  2x{\text{ }}x > 3 \\
  {x^2}{\text{ }}1 < x \leqslant 3 \\
  3x{\text{ }}x \leqslant 1 \\
\end{gathered} \right\}$
In $f\left( { - 1} \right)$, x = -1 which lies in the interval $x \leqslant 1$ and for this interval the function is defined as $f\left( x \right) = 3x$
By putting x = -1 in the above function, we get
$
  f\left( { - 1} \right) = 3\left( { - 1} \right) \\
   \Rightarrow f\left( { - 1} \right) = - 3{\text{ }} \to {\text{(1)}} \\
 $
In $f\left( 2 \right)$, x = 2 which lies in the interval $1 < x \leqslant 3$ and for this interval the function is defined as $f\left( x \right) = {x^2}$
By putting x = 2 in the above function, we get
$
  f\left( 2 \right) = {2^2} \\
   \Rightarrow f\left( 2 \right) = 4{\text{ }} \to {\text{(2)}} \\
 $
In $f\left( 4 \right)$, x = 4 which lies in the interval $x > 3$ and for this interval the function is defined as $f\left( x \right) = 2x$
By putting x = 4 in the above function, we get
$
  f\left( 4 \right) = 2 \times 4 \\
   \Rightarrow f\left( 4 \right) = 8{\text{ }} \to {\text{(3)}} \\
 $
The value of the expression $f\left( { - 1} \right) + f\left( 2 \right) + f\left( 4 \right)$ can be obtained by using equation
$
  f\left( { - 1} \right) + f\left( 2 \right) + f\left( 4 \right) = - 3 + 4 + 8 \\
   \Rightarrow f\left( { - 1} \right) + f\left( 2 \right) + f\left( 4 \right) = 9 \\
 $
Therefore, the value of $f\left( { - 1} \right) + f\left( 2 \right) + f\left( 4 \right)$ is 9.
Hence, option A is correct.

Note- In this particular problem, the given function has three different definitions according to the three intervals. Here, the value of the function for any specific value of x can be calculated by considering the corresponding definition of the function and then substituting that particular value of the variable x.