Mr. Integral
Math notes/Pre-Calculus/Functions Review

Domain, Codomain and Range — Cheat sheet

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Key method

For f:A→Bf:A\to B, the range is f(A)={f(x):x∈A}⊆Bf(A)=\{f(x):x\in A\}\subseteq B. Restricting the domain can shrink the range. Enlarging the codomain alone does not change any computed value.

Example

For f:[−2,3]→Rf:[-2,3]\to\mathbb R, f(x)=x2f(x)=x^2, the minimum is 00 at x=0x=0 and the maximum is 99 at x=3x=3. The range is [0,9][0,9], not [4,9][4,9]: squaring only the endpoints misses the interior minimum. With domain [1,3][1,3], the same rule has range [1,9][1,9].

Avoid this mistake

The codomain cannot be inferred uniquely from the formula. State it explicitly when asking whether a function is onto.