Range

Algebra Level 4

Range of the function f ( x ) = x 1 + x f(x)=\frac{x}{1+x} where x {x} denotes the fractional part function is [ 0 , a b ) [0,\frac{a}{b}) and g c d ( a , b ) = 1 gcd(a,b)=1 .Then value of a × b a\times b is equal to

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The answer is 2.

Akshay Sharma by Akshay Sharma , 21, India

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1 solution

Rishabh Jain
Mar 16, 2016

Function can be written as:- f ( x ) = 1 − 1 1 + { x } f(x)=1-\dfrac{1}{1+\{x\} }

0 ≤ { x } ≤ 1 ⟹ 1 ≤ { x } + 1 ≤ 2 ⟹ 1 2 ≤ 1 { x } + 1 ≤ 1 ⟹ − 1 ≤ − 1 { x } + 1 ≤ − 1 2 0 ≤ 1 − 1 { x } + 1 = f ( x ) ≤ 1 2 \begin{aligned}&0\leq\{x\}\leq 1\\&\implies 1\leq\{x\}+1\leq 2\\&\implies \dfrac 12\leq\dfrac{1}{\{x\}+1}\leq 1\\&\implies -1\leq-\dfrac{1}{\{x\}+1}\leq \dfrac{-1}{2}\\&0\leq1-\dfrac{1}{\{x\}+1}=f(x)\leq \dfrac 12\end{aligned} Hence, 1 × 2 = 2 \Huge 1\times 2=\boxed 2

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