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4.Invertible Functions
Class 11 • Mathematics • NCERT
Invertible Functions
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This lesson explains invertible functions as prescribed by NCERT Class 11. Students learn when a function has an inverse and how one-one and onto properties are essential for invertibility.
Lesson Objectives
- Understand inverse functions.
- Learn conditions for invertibility.
- Find inverse of simple functions.
- Answer NCERT exam questions confidently.
1. Meaning of Inverse Function
If a function f maps elements from set A to set B, then its inverse function f⁻¹ maps elements from B back to A.
If f(a) = b, then f⁻¹(b) = a
2. Condition for Invertibility
A function is invertible if and only if it is both one-one and onto.
Invertible ⇔ One-one + Onto
3. Verification Using Composition
If f and f⁻¹ exist, then:
f ∘ f⁻¹ = I and f⁻¹ ∘ f = I
where I(x) = x is the identity function.
4. Simple NCERT Examples
Let f(x) = 2x + 3, f : ℝ → ℝ
Replace f(x) = y → y = 2x + 3
Solve for x → x = (y − 3)/2
∴ f⁻¹(x) = (x − 3)/2
5. Functions That Are Not Invertible
Functions that are not one-one or not onto do not have inverses.
Example: f(x) = x² on ℝ (not one-one, not invertible)
Practice Questions (NCERT)
- Define inverse of a function.
- What condition must a function satisfy to be invertible?
- Is every one-one function invertible?
- Is every onto function invertible?
- Find the inverse of f(x) = x + 5.
- Find the inverse of f(x) = 3x.
- Why does f(x) = x² not have an inverse on ℝ?
- State the identity function.
- What is f ∘ f⁻¹ equal to?
- Is this topic part of NCERT Class 11 syllabus?
✅ Show Answer Key
- Function reversing the mapping of f
- Must be one-one and onto
- No
- No
- f⁻¹(x) = x − 5
- f⁻¹(x) = x/3
- Not one-one
- I(x) = x
- Identity function
- Yes
© Aviate Learning – Relations & Functions (NCERT Class 11)