Question

Let $A = \{ 9,10,11,12,13\} $ and $f:A \to N$ be defined by \[f\left( n \right)\]=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 1

Hint: Consider the set is given $A = \{ 9,10,11,12,13\} $ and the function is defined as $f:A \to N$ and its range \[f\left( n \right)\] contain the highest prime factor of $n$.
First, find the factors of elements of the set $A = \{ 9,10,11,12,13\} $.
Find the highest prime factor of each element.
The set of the highest prime factors of the individual elements is the range \[f\left( n \right)\]

Complete step-by-step solution:
The function is defined $f:A \to N$ where, $A = \{ 9,10,11,12,13\} $.
Take the values of $n = \{ 9,10,11,12,13\} $, find the factor of each element.
The factors of $9$ =$3 \times 3$
The factors of $10$ =$2 \times 5$
The factors of $11$ =$1 \times 11$
The factors of $12$ =$2 \times 2 \times 3$
The factors of $13$ =$1 \times 13$
\[f\left( n \right)\]=highest prime factor of $n$
\[f\left( 9 \right)\]=the highest prime factor of $9$
$ \Rightarrow $ \[f\left( 9 \right) = 3\]
\[f\left( {10} \right)\]=the highest prime factor of $10$
$ \Rightarrow $ \[f\left( {10} \right) = 5\]
\[f\left( {11} \right)\]=the highest prime factor of $11$
$ \Rightarrow $ \[f\left( {11} \right) = 11\]
\[f\left( {12} \right)\]=the highest prime factor of $12$
$ \Rightarrow $ \[f\left( {12} \right) = 3\]
\[f\left( {13} \right)\]=the highest prime factor of $13$
$ \Rightarrow $ \[f\left( {13} \right) = 13\]
The range of $f$ is the set of all $f(n)$ =$\left\{ {3,5,11,13} \right\}$
The range \[f\left( n \right)\] is $\left\{ {3,5,11,13} \right\}$.

Option B is the correct answer.

Note: Here are some basic ideas about the domain and range of a function.
 Domain: The set of possible input values, the values go into a function is called domain.
Range: The set of possible output values and all the values that come out is called range.
For example: If the set $A = \left\{ {1,2,3} \right\}$ and $f(x) = {x^2}$ then,
$f(1) = {1^2}$
$ \Rightarrow f(1) = 1$
$f(2) = {2^2}$
$ \Rightarrow f(2) = 4$
$f(3) = {3^2}$
$ \Rightarrow f(3) = 9$
The range is set of all elements $\left\{ {1,4,9} \right\}$ and domain is $\left\{ {1,2,3} \right\}$.