Question & Answer
QUESTION

Let $A=\left\{ 9,10,11,12,13 \right\}$ and $f:A\to N$ be defined by $f\left( n \right)=$ the highest prime factor of $n$, then its range is.
(a) $\left\{ 13 \right\}$
(b) $\left\{ 3,5,11,13 \right\}$
(c) $\left\{ 11,13 \right\}$
(d) $\left\{ 2,3,5,11 \right\}$

ANSWER Verified Verified
Hint: For solving this question we should know how to find the factors of any number. Moreover, some basic details about the range of any function will be needed.

Complete step-by-step answer:

Given:
$A=\left\{ 9,10,11,12,13 \right\}$ and $f:A\to N$

Domain for any function is defined as the set of input values for which the function is valid.
From the above data, we can say that input value $\left( n \right)$ in the function will be from $A=\left\{ 9,10,11,12,13 \right\}$ or the domain of the given function will be $A=\left\{ 9,10,11,12,13 \right\}$ .

It is given that, $f\left( n \right)=$ highest prime factor of $n$ . So, for any input value, $\left( n \right)$ the output value is the highest prime factor of $n$ .
Thus, $n$ can take values $9,10,11,12,13$ . First, we will find the highest prime factor corresponding to each value of $n$ .

Before finding the highest prime factor we should know that, what do we mean by prime number.

A prime number is a natural number greater than 1 such that it cannot be divisible by any other number except 1 and by itself. For example 2, 3, 5, 7, 11, 13, 17 are prime numbers.
Highest prime factor of 9.

We can write, $9=1\times 3\times 3$ . Then, we can say that the factors of 9 are $1,3,9$ . So, the highest prime factor of 9 will be 3.
Then,
$f\left( 9 \right)=3..........\left( 1 \right)$

Highest prime factor of 10.
We can write, $10=1\times 2\times 5$ . Then, we can say that the factors of 10 are $1,2,5,10$ . So, the highest prime factor of 10 will be 5.

Then,
$f\left( 10 \right)=5..........\left( 2 \right)$
Highest prime factor of 11.

We can write, $11=1\times 11$ . Then, we can say that the factors of 11 are $1,11$ . So, the highest prime factor of 11 will be 11 itself.
Then,
$f\left( 11 \right)=11..........\left( 3 \right)$
Highest prime factor of 12.

We can write, $12=1\times 2\times 2\times 3$ . Then, we can say that the factors of 12 are $1,2,3,4,6,12$ . So, the highest prime factor of 12 will be 3.
Then,
$f\left( 12 \right)=3.........\left( 4 \right)$
Highest prime factor of 13.

We can write, $13=1\times 13$ . Then, we can say that the factors of 13 are $1,13$ . So, the highest prime factor of 13 will be 13 itself.
Then,
$f\left( 13 \right)=13..........\left( 5 \right)$

From $\left( 1 \right),\left( 2 \right),\left( 3 \right),\left( 4 \right)$ and $\left( 5 \right)$ , we have:
$\begin{align}
  & f\left( 9 \right)=3 \\
 & f\left( 10 \right)=5 \\
 & f\left( 11 \right)=11 \\
 & f\left( 12 \right)=3 \\
 & f\left( 13 \right)=13 \\
\end{align}$
Now, from above we can say that the set of output values of the given function will be $\left\{ 3,5,11,13 \right\}$ .

We know that the range of any function is the set of output values of that function for the valid input values or its domain.

Hence, the range will be $\left\{ 3,5,11,13 \right\}$ .
Thus, (b) is the correct option.

Note: Here, students should not confuse the range of the function and understand the symbols mentioned in the question properly. And one should remember the word highest mentioned in the question while calculating the output value of the given function.