Domain & Range ExplorerJEE Main

Where a function lives, and what it hits — interactive Mathematics simulation for IIT-JEE.

Concept

Domain = every x the formula tolerates (no √ of negatives, no ÷0, no log of non-positives). Range = every y actually achieved. A function is one-one if no horizontal line hits it twice — the prerequisite for having an inverse. Domain-range analysis is the opening move of half of JEE algebra & calculus.

Key formula

\sqrt{g(x)}: g \ge 0; \quad \frac{1}{g(x)}: g \ne 0; \quad \log g(x): g > 0

Derivation

Domain: intersect all constraints. Range strategies: (1) solve y = f(x) for x and demand x-existence; (2) complete the square for quadratics; (3) monotonic pieces map endpoints to endpoints.

Example: y = x² − 4x + 3 = (x−2)² − 1 ≥ −1 → range [−1, ∞). Sigmoid: eˣ/(1+eˣ) is squeezed in (0,1), strictly increasing → invertible with inverse ln(y/(1−y)).

Scenarios to explore

Domain & Range Explorer — Where a function lives, and what values it actually hits.

Real-world applications

Composite functions: range of the inner must fit the domain of the outer.
Inverse trig conventions are forced domain restrictions.
Quadratic range questions & max-min without calculus.

JEE exam tips

For y = (ax+b)/(cx+d): range is ℝ \ {a/c} — the horizontal asymptote is missed.
|x−a| + |x−b| has minimum |a−b|, achieved on the whole segment [a,b].
Even functions are never one-one on symmetric domains; restrict to invert.

Common mistakes

Range of x² as ℝ (it's [0, ∞)).
Ignoring points where a simplified formula differs from the original ((x²−1)/(x−1) has a hole at 1).
Assuming onto without checking the codomain.

Exam traps to avoid

Domain of √(log x) is x ≥ 1, not x > 0 — chain the conditions.
f(x) = x²: one-one on [0, ∞), NOT on ℝ — one-one-ness depends on the domain, not the formula.