Question

Let A$ = \{ 12,13,14,15,16,17\} $ and $f:$ A $ \to $ Z be a function given by $f\left( x \right) = $ highest prime factor of $x$. Find range of $f$.

Answer 1

Hint: The given function \[f\left( x \right)\] is defined over domain A $ = \{ 12,13,14,15,16,17\} $ and we have to find the range of $f\left( x \right)$ . the function $f\left( x \right) = $ highest prime factor of $x$. Firstly, find the factor of each number present in its domain that is from $12$ to $17$, then choose the factor which is highest among all other factors of a number and grouped together which is the required range of the function $f\left( x \right)$.

Complete step-by-step answer:
Given, $f:$ A $ \to $Z be a function such that $f\left( x \right) = $ highest prime factor of $x$.
Domain A $ = \{ 12,13,14,15,16,17\} $.
Now, we have to write the prime factors of each number.
Prime factor of $12 = 2 \times 2 \times 3$
The highest prime factor of $12$ is $3$.
Prime factor of $13 = 13$
The highest prime factor of $13$ is $13$.
Prime factor of $14 = 2 \times 7$
The highest prime factor of $14$ is $7$.
Prime factor of $15 = 3 \times 5$
The highest prime factor of $15$ is $5$.
Prime factor of $16 = 2 \times 2 \times 2 \times 2$
The highest prime factor of $16$ is $2$.
Prime factor of $17 = 17$
The highest prime factor of $17$ is $17$.
So, the highest prime factor of numbers in the domain A is $\left\{ {3,13,7,5,2,17} \right\}$. Now, putting them in sequence we get,

The range of the given function $f\left( x \right)$ is $\left\{ {2,3,5,7,13,17} \right\}$.

Note:
The domain of a function is the complete set of possible values of the independent variable (usually $x$). The range of a function is the complete set of all possible resulting values of the dependent variable (usually $y$) after we have substituted the domain.