Class 12 Mathematics - Chapter Relations and Functions NCERT Solutions | Consider f : R+ → [4, ∞
Consider f : R+ → [4, ∞) given by f(x) = x2 + 4. Show that f is invertible with the inverse f–1 of f given by 
, where R+ is the set of all non-negative real numbers.
f : R+ → [4, ∞) is given as f(x) = x2 + 4.
One-one:
Let f(x) = f(y).
∴ f is a one-one function.
Onto:
For y ∈ [4, ∞), let y = x2 + 4.
Therefore, for any y ∈ R, there exists 
such that
.
∴ f is onto.
Thus, f is one-one and onto and therefore, f - 1 exists.
Let us define g: [4, ∞) → R+ by,
∴
Hence, f is invertible and the inverse of f is given by
More Questions From Class 12 Mathematics - Chapter Relations and Functions
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(i) Symmetric but neither reflexive nor transitive.
(ii) Transitive but neither reflexive nor symmetric.
(iii) Reflexive and symmetric but not transitive.
(iv) Reflexive and transitive but not symmetric.
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(i) Relation R in the set A = {1, 2, 3,13, 14} defined as
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(ii) Relation R in the set N of natural numbers defined as
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(iii) Relation R in the set A = {1, 2, 3, 4, 5, 6} as
R = {(x, y): y is divisible by x}
(iv) Relation R in the set Z of all integers defined as
R = {(x, y): x − y is as integer}
(v) Relation R in the set A of human beings in a town at a particular time given by
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(b) R = {(x, y): x and y live in the same locality}
(c) R = {(x, y): x is exactly 7 cm taller than y}
(d) R = {(x, y): x is wife of y}
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- Q:- Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as
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- Q:- Show that the relation R in R defined as R = {(a, b): a ≤ b}, is reflexive and transitive but not symmetric.
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