Functions

Piecewise Definition:
- For $x \geq 0$ , $f (x) = \frac{x}{1 + x}$ . As $x \to \infty, f (x) \to 1$ . Range $= [0, 1)$ .
- For $x < 0$ , $f (x) = \frac{x}{1 - x}$ . As $x \to - \infty, f (x) \to - 1$ . Range $= (- 1, 0)$ .
Combine Domains: The total range is the union of $(- 1, 0) \cup [0, 1) = (- 1, 1)$ .
Boundaries Analysis: Note that $\frac{x}{1 + ∣ x ∣}$ can never exactly equal $1$ because $∣ x ∣ < 1 + ∣ x ∣$ for all finite $x$ . Similarly for $- 1$ . Thus, the interval is open.

This hard 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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Solution

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