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MathsMediumClass 11
Composition & Inverse Functions
Sets, Relations & Functions
10
JEE Qs
8%
Hard
90
min
Always verify the bijectivity of a function (one-one and onto) before attempting to find its inverse, and carefully define the domain and range of the inverse function.
🧮 Key Formulas
(g o f)(x) = g(f(x))
A function f: A -> B has an inverse f⁻¹: B -> A if and only if f is bijective (one-one and onto).
If y = f(x), then x = f⁻¹(y).
(f o f⁻¹)(x) = x for all x in Domain(f⁻¹)
(f⁻¹ o f)(x) = x for all x in Domain(f)
(g o f)⁻¹(x) = (f⁻¹ o g⁻¹)(x)
✅ Key Points for JEE
- 1For (g o f)(x) to be defined, the range of f must be a subset of the domain of g.
- 2An inverse function f⁻¹ exists if and only if f is both one-one (injective) and onto (surjective), meaning f is a bijection. If a function is not bijective over its entire domain, its inverse may exist over a restricted domain.
- 3The domain of f becomes the range of f⁻¹, and the range of f becomes the domain of f⁻¹.
- 4The graph of y = f⁻¹(x) is always the reflection of the graph of y = f(x) about the line y = x.
- 5To find the inverse of y = f(x), solve for x in terms of y, then swap x and y, and specify the domain and range of the resulting inverse function.
⚠️ Common Mistakes
- ✕Assuming (g o f)(x) is always defined without checking if the range of f is a subset of the domain of g.
- ✕Incorrectly defining the domain and range of the inverse function, especially when the original function's domain was restricted to ensure bijectivity.
- ✕Confusing the inverse function f⁻¹(x) with the reciprocal 1/f(x).
- ✕Algebraic errors when rearranging y = f(x) to solve for x in terms of y.
NCERT Chapters
- Class 11 Mathematics Ch 2: Relations and Functions