Question

If $R\to R$ is defined by $f\left( x \right)=\left\{ \begin{matrix}
   x+4 & for\text{ }x<-4 \\
   3x+2 & for\text{ }-4\le x<4 \\
   x-4 & for\text{ }x\ge 4 \\
\end{matrix} \right\}$ . Then the correct matching of List I from List II is:

List I	List II
A. $f\left( -5 \right)+f\left( -4 \right)$	1. 14
B. $f\left( \left| f\left( -8 \right) \right| \right)$	2. 4
C. $f\left[ f\left( -7 \right)+f\left( 3 \right) \right]$	3. -11
D. $f\left[ f\left\{ f\left( 0 \right) \right\} \right]+1$	4. -1
	5.1
	6.0

1) A=3, B=6, C=2, D=5
2) A=3, B=4, C=2, D=5
3) A=4, B=3, C=2, D=1
4) A=3, B=6, C=5, D=2

Answer 1

Hint: Here in this question, we have been asked to match the elements in List I with the ones in List II. Given that $f\left( x \right)=\left\{ \begin{matrix}
   x+4 & for\text{ }x<-4 \\
   3x+2 & for\text{ }-4\le x<4 \\
   x-4 & for\text{ }x\ge 4 \\
\end{matrix} \right\}$ . For answering this question we will substitute the values for x accordingly and verify.

Complete step-by-step solution:
Now considering the question, we have been asked to match the elements in List I with the ones in List II. Given that $f\left( x \right)=\left\{ \begin{matrix}
   x+4 & for\text{ }x<-4 \\
   3x+2 & for\text{ }-4\le x<4 \\
   x-4 & for\text{ }x\ge 4 \\
\end{matrix} \right\}$.
For $f\left( -5 \right)+f\left( -4 \right)$ , let us substitute $x=-5$ in $f\left( x \right)$ . By doing that we will have $\Rightarrow x+4=\left( -5 \right)+4=-1$ and similarly $f\left( -4 \right)=3\left( -4 \right)+2=-10$ .
Hence $f\left( -5 \right)+f\left( -4 \right)=-11$ . Therefore “A” should be matched to “3”.
For $f\left( -8 \right)$ , let us substitute $x=-8$ in $f\left( x \right)$ . By doing that we will have $\Rightarrow -8+4=-4$.
Now we can say that $\left| f\left( -8 \right) \right|=4$ . By further evaluating it we can say that $f\left( \left| f\left( -8 \right) \right| \right)=f\left( 4 \right)=0$ .
Therefore “B” should be matched to “6”.
For $f\left[ f\left( -7 \right)+f\left( 3 \right) \right]$ , let us substitute$x=-7$ in $f\left( x \right)$ . By doing that we will have $\Rightarrow x+4=\left( -7 \right)+4=-3$ and similarly $f\left( 3 \right)=3\left( 3 \right)+2=11$ .
Hence $f\left( -7 \right)+f\left( 3 \right)=8$ .
Now $f\left[ f\left( -7 \right)+f\left( 3 \right) \right]=f\left( 8 \right)\Rightarrow 8-4=4$.
Therefore “C” should be matched to “2”.
Now let us evaluate $f\left( 0 \right)$ , we can say that $f\left( 0 \right)=3\left( 0 \right)+2=2$ . Now we will evaluate $f\left\{ f\left( 0 \right) \right\}=f\left( 2 \right)=3\left( 2 \right)+2=8$ . Now we will evaluate $f\left[ f\left\{ f\left( 0 \right) \right\} \right]=f\left[ 8 \right]=8-4=4$ .
Now we will have $f\left[ f\left\{ f\left( f\left( 0 \right) \right) \right\} \right]=f\left[ 4 \right]=4-4=0$ .
Now we can say that $f\left[ f\left\{ f\left( f\left( 0 \right) \right) \right\} \right]+1$ is 1.
Therefore “D” should be matched to “5”.

Note: While answering questions of this type we should be sure with the simplifications that we are going to perform in between the steps. Let us consider that if someone has missed the conditions given then they will end up having a wrong answer.