Verified Solution • 2025–26 Session moderate

QJEEM09Madd9c: JEE Main 2009 Mathematics - Class 11 JEEM09Madd9c Solution: For real x, let f(x) = x 3 + 5x + 1, then

For real x, let f(x) = x3 + 5x + 1, then

Multiple Choice Options

  • A f is one-one but not onto R
  • B f is onto R but not one-one
  • C f is one-one and onto R
  • D f is neither one-one nor onto R

Step-by-Step Verified Solution

Given that f(x)=x3+5x+1f\left( x \right) = {x^3} + 5x + 1

∴\therefore      \,\,\,\,\, f'\left( x \right) = 3{x^2} + 5 > 0,\,\,\,\forall x \in R

⇒f(x)  \Rightarrow f\left( x \right)\,\, is strictly increasing on RR

⇒f(x)\Rightarrow f\left( x \right) is one one

∴\therefore        \,\,\,\,\,\,\, Being a polynomial f(x)f(x) is cont. and inc.

on RR with lim⁡x→∞ f(x)=−∞\mathop {\lim }\limits_{x \to \infty } \,f\left( x \right) = - \infty

and lim⁡x→∞ f(x)=∞\mathop {\lim }\limits_{x \to \infty } \,f\left( x \right) = \infty

∴\therefore        \,\,\,\,\,\,\, Range of f=(−∞,∞)=Rf = \left( { - \infty ,\infty } \right) = R

Hence ff is onto also, So, ff is one and onto R.R.

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