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.