Let $f:R \to R$ be a function defined by
$$f(x) = {\left( {2\left( {1 - {{{x^{25}}} \over 2}} \right)(2 + {x^{25}})} \right)^{{1 \over {50}}}}$$. If the function $g(x) = f(f(f(x))) + f(f(x))$, then the greatest integer less than or equal to g(1) is ____________.
Answer (integer)
2
Solution
<p>Given,</p>
<p>$$f(x) = {\left( {2\left( {1 - {{{x^{25}}} \over 2}} \right)\left( {2 + {x^{25}}} \right)} \right)^{{1 \over {50}}}}$$</p>
<p>and $$g(x) = f\left( {f\left( {f\left( x \right)} \right)} \right) + f\left( {f\left( x \right)} \right)$$</p>
<p>$\therefore$ $$g(1) = f\left( {f\left( {f\left( 1 \right)} \right)} \right) + f\left( {f\left( 1 \right)} \right)$$</p>
<p>Now, $$f(1) = {\left( {2\left( {1 - {{{1^{25}}} \over 2}} \right)\left( {2 + {1^{25}}} \right)} \right)^{{1 \over {50}}}}$$</p>
<p>$$ = {\left( {2\left( {1 - {1 \over 2}} \right)\left( {2 + 1} \right)} \right)^{{1 \over {50}}}}$$</p>
<p>$= {\left( 3 \right)^{{1 \over {50}}}}$</p>
<p>$\therefore$ $f\left( {f\left( 1 \right)} \right) = f\left( {{3^{{1 \over {50}}}}} \right)$</p>
<p>$$ = {\left( {2\left( {1 - {{{{\left( {{3^{{1 \over {50}}}}} \right)}^{25}}} \over 2}} \right)\left( {2 + {{\left( {{3^{{1 \over {50}}}}} \right)}^{25}}} \right)} \right)^{{1 \over {50}}}}$$</p>
<p>$$ = {\left( {2\left( {1 - {{{3^{{1 \over 2}}}} \over 2}} \right)\left( {2 + {3^{{1 \over 2}}}} \right)} \right)^{{1 \over {50}}}}$$</p>
<p>$$ = {\left( {2 \times \left( {{{2 - \sqrt 3 } \over 2}} \right)\left( {2 + \sqrt 3 } \right)} \right)^{{1 \over {50}}}}$$</p>
<p>$$ = {\left[ {\left( {2 - \sqrt 3 } \right)\left( {2 + \sqrt 3 } \right)} \right]^{{1 \over {50}}}}$$</p>
<p>$= {\left( {4 - 3} \right)^{{1 \over {50}}}}$</p>
<p>$= {1^{{1 \over {50}}}} = 1$</p>
<p>Now, $$f\left( {f\left( {f\left( 1 \right)} \right)} \right) = f(1) = {3^{{1 \over {50}}}}$$</p>
<p>$\therefore$ $$g(1) = f\left( {f\left( {f\left( 1 \right)} \right)} \right) + f\left( {f\left( 1 \right)} \right)$$</p>
<p>$= {3^{{1 \over {50}}}} + 1$</p>
<p>Now, greatest integer less than or equal to $g(1)$</p>
<p>$= \left[ {g(1)} \right]$</p>
<p>$= \left[ {{3^{{1 \over {50}}}} + 1} \right]$</p>
<p>$= \left[ {{3^{{1 \over {50}}}}} \right] + \left[ 1 \right]$</p>
<p>$= [1.02] + 1$</p>
<p>$= 1 + 1 = 2$</p>
About this question
Subject: Mathematics · Chapter: Sets, Relations and Functions · Topic: Sets and Operations
This question is part of PrepWiser's free JEE Main question bank. 195 more solved questions on Sets, Relations and Functions are available — start with the harder ones if your accuracy is >70%.