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

Hint: 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 \[q \ne 0\].