Domain and Range of a Real Function
Domain: Generally, real functions in calculus are described by some formula and their domain are not explicitly stated. Find the domain of the function f(say) we find all real numbers x for which f(x) is a real valued. In other word, determining the domain of a function means finding all real numbers x for which f(x) is real
Example: If f(x) = \(\sqrt{2-x}\), then f(x) is real for x ≤ 2. For x > 2, f(x) is not real. So, domain of f(x) is the set of all real numbers less than or equal to 2 i.e., (-∞, 2]
Rules for Finding Domain:
(i) Expression under even root (i.e., square root, fourth root) ≥ 0.
(ii) Denominator ≠ 0.
(iii) If domain of y = f(x) and y = g(x) are D₁ and D₂ respectively, then the domain of f(x) ± g(x) or f(x). g(x) is D₁ Ç D₂, while domain of f(x)/g(x) is D₁ Ç D₂ – {g(x) = 0}.
Example: The domain of the function \(f(x)=\sqrt{\frac{(x+1)(x-3)}{(x-2)}}\).
Solution: Given that \(f(x)=\sqrt{\frac{(x+1)(x-3)}{(x-2)}}\).
\(f(x)=\sqrt{\frac{(x+1)(x-3)}{(x-2)}\times \frac{(x-2)}{(x-2)}}\).
\(f(x)=\frac{\sqrt{(x+1)(x-3)(x-2)}}{(x-2)}\).
(x + 1) (x – 3) (x – 2) ≥ 0
{(x – (-1)) (x – 3) (x – 2)} ≥ 0
-1 ≤ x ≤ 2 or x ≥ 0
x-2 ≠ 0 ⇒ x ≠ 2
Domain of f = {[-1, 2] ∪ [3, ∞)} – {2}
= [-1, 2) ∪ [3, ∞)
Codomain: The set of all possible outcomes for the function f is called the codomain if the function f.
Example: Let f: [a, b, c] → [α, β, γ, δ] is a function, that the codomain of f is [α, β, γ, δ].
Range: The range of a function f(x) is the set of values of f(x) which it attains at points in its domain. For a real function the codomain is always a subset of R. so, range of a real function f is the set of all points y such that y = f(x).
Rules for Finding Range:
First of all, find the domain of y = f(x)
(i) If domain belongs to finite number of points
⇒ range belongs to set of corresponding f(x) values.
(ii) If domain belongs to R or R – [Some finite points] then, express x in terms of y. from this find y for x to be defined (i.e., find the value of y for which x exists).
(iii) If domain belongs a finite interval, find the least and greatest value for range using monotonicity.