Functions

One-one (Injection): $f^{'} (x) = \frac{( 1 + x ) - x}{( 1 + x ) ^{2}} = \frac{1}{( 1 + x ) ^{2}}$ . Since the derivative is strictly positive for $x \in [0, \infty)$ , the function is strictly increasing and thus one-one.
Range Analysis: As $x \to \infty$ , $f (x) \to \frac{x}{x} = 1$ . Since $x$ is always positive, $f (x)$ is always less than 1. Thus, Range $= [0, 1)$ .
Onto (Surjection): The codomain is provided as $[0, \infty)$ . Since the range ( $[0, 1)$ ) is a proper subset of the codomain, the function is not onto.
Conclusion: One-one but not onto.

This medium difficulty mathematics question is from the chapter sets, relations and functions, covering the topic of functions. It appeared in the 2025 exam.

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Given $f (x) = x^{1/ x}$ for $x > 0$ . The maximum value of $f (x)$ is: