Question

If the function $f:N \to N$ is defined by \left[ {\dfrac{{f\left( {25} \right)}}{{\left\\{ {f\left( {16} \right) + f\left( 1 \right)} \right\\}}}} \right] is equal to
\left( 1 \right)$$$\dfrac{5}{6}$ \left( 2 \right)\dfrac{5}{7}$ $$\left( 3 \right)\dfrac{5}{3} $$\left( 4 \right)$$$1

Answer 1

We have to find the value of $\;\left[ {f\left( {25} \right)/f\left( {16} \right) + f\left( 1 \right)} \right]$. We solve this question using the concept of values of the root of numbers . We define the function $f\left( x \right)$ by its value and then by putting the value of $x$ in the function we get the value of the function then by simplifying we get the value of the required function .

Complete step-by-step solution:
Given : $f\left( x \right) = \sqrt x$
We have to find the value of $\;\left[ {f\left( {25} \right)/f\left( {16} \right) + f\left( 1 \right)} \right]$
We have to find the value of the function $f\left( x \right)$when $x = 25,x = 16$and $x = 1.$
We get the values of the function by putting the values at these values of $x$ and then put the value of the function in the expression for which we have to find the value .
$f\left( x \right) = \sqrt x$
At $x = 25$
$f(25) = \sqrt {25}$
We know that the square root of a value is the product of a term which is multiplied by itself to give the number whose root we have to find .
So ,
$f\left( {25} \right) = 5$
Similarly ,
At $x = 16$
$f(16) = \sqrt {16}$
$f\left( {16} \right) = 4$
Similarly ,
At $x = 1$
$f(1) = \sqrt 1$
$f\left( 1 \right) = 1$
Putting the values of $f\left( {25} \right),f\left( {16} \right)$and $f\left( 1 \right)$in the expression , we get
\left[ {\dfrac{{f\left( {25} \right)}}{{\left\\{ {f\left( {16} \right) + f\left( 1 \right)} \right\\}}}} \right] = \left[ {{\text{ }}\dfrac{5}{{\left( {4 + 1} \right)}}} \right]
\;\left[ {\dfrac{{f\left( {25} \right)}}{{\left\\{ {f\left( {16} \right) + f\left( 1 \right)} \right\\}}}} \right] = \dfrac{5}{5}
\;\left[ {{\text{ }}\dfrac{{f\left( {25} \right)}}{{\left\\{ {f\left( {16} \right) + f\left( 1 \right)} \right\\}}}} \right] = 1
Thus the value of \left[ {{\text{ }}\dfrac{{f\left( {25} \right)}}{{\left\\{ {f\left( {16} \right) + f\left( 1 \right)} \right\\}}}} \right]is equal to $1$
Hence , the correct option is $\left( 4 \right)$.

Note: A square root number can be a rational or irrational number . If the square root of a number can be represented in terms of natural numbers then it is a rational number i.e. it can be written in the form of $\dfrac{p}{q}$ where $q \ne 0$. And if the square root of a number can not be represented in terms of natural number than it is an irrational number i.e. it can be written in the form of $\dfrac{p}{q}$ where $q \ne 0$.