Physics and Mathematics
Classification of Functions
1. Concept Overview
A function can be classified in different ways based on:
- its rule of mapping,
- its nature of outputs, and
- its graphical behavior.
This classification helps us:
- understand functions better,
- apply correct theorems,
- and solve problems in calculus easily.
2. Main Types of Functions (Based on Mapping)
(a) One–One Function (Injective Function)
A function is called one–one if:
Different inputs give different outputs.
Mathematically:
If [f(x₁)=f(x₂)], then [x₁=x₂].
Example:
[f(x)=2x]
(b) Many–One Function
A function is called many–one if:
Two or more different inputs give the same output.
Example:
[f(x)=x²], since [f(2)=f(−2)]
(c) Onto Function (Surjective Function)
A function is called onto if:
Every element of the codomain is an image of at least one element of the domain.
That is:
Range = Codomain
(d) Into Function
A function is called into if:
At least one element of the codomain is not an image of any element of the domain.
That is:
Range ⊂ Codomain
3. Algebraic Classification of Functions
(a) Polynomial Function
A function of the form:
[f(x)=aₙxⁿ+…+a₁x+a₀]
Example:
[f(x)=x³−2x+1]
(b) Rational Function
A function expressed as a ratio of two polynomials.
Example:
[f(x)=\dfrac{x+1}{x−2}]
(c) Irrational Function
A function containing variables inside a root.
Example:
[f(x)=\sqrt{x+3}]
(d) Transcendental Functions
Functions that are not algebraic.
Examples:
- Exponential: [f(x)=e^x]
- Logarithmic: [f(x)=\log x]
- Trigonometric: [f(x)=\sin x]
4. Special Types of Functions
(a) Identity Function
A function that maps each element to itself.
[f(x)=x]
(b) Constant Function
A function that gives the same output for all inputs.
[f(x)=c]
(c) Modulus Function
[f(x)=|x|]
(d) Greatest Integer Function
[f(x)={[}x{]}]
5. Conceptual Questions with Solutions
1. What is a one–one function?
A one–one function is a function in which different inputs produce different outputs, ensuring uniqueness of mapping.
2. Can a many–one function be a valid function?
Yes. A many–one function is a valid function as long as each input has exactly one output.
3. What does an onto function signify?
An onto function signifies that the range of the function is equal to the codomain.
4. Why is every onto function not necessarily one–one?
Because an onto function may map multiple inputs to the same output.
5. Why is the identity function always one–one?
Because each input is mapped to itself, ensuring distinct outputs for distinct inputs.
6. Is a constant function one–one?
No. A constant function maps all inputs to the same output, so it is many–one.
7. Can a function be both one–one and onto?
Yes. Such a function is called a bijective function.
8. Why is bijective function important?
A bijective function is important because it guarantees the existence of inverse function.
9. Is every polynomial function one–one?
No. Some polynomial functions may be many–one.
10. Why are transcendental functions different?
Transcendental functions cannot be expressed using algebraic operations alone.
11. What type of function is [f(x)=|x|]?
It is a modulus function and also a many–one function.
12. Why is the greatest integer function discontinuous?
Because it shows sudden jumps at integer points.
13. Can a function be into and one–one at the same time?
Yes. A function can be into and one–one simultaneously.
14. Is every onto function invertible?
No. Only bijective functions are invertible.
15. Why is classification of functions necessary?
Because classification helps in applying correct properties and theorems in calculus.
6. FAQ / Common Misconceptions
1. Every function is one–one.
False. Many functions are many–one.
2. Onto and one–one mean the same.
False. Onto refers to codomain coverage, while one–one refers to uniqueness of outputs.
3. A constant function can be one–one.
False. A constant function is always many–one.
4. Every function has an inverse.
False. Only bijective functions have inverse functions.
5. Polynomial functions are always continuous.
True. All polynomial functions are continuous everywhere.
6. Modulus function is one–one.
False. It is a many–one function.
7. Identity function is many–one.
False. Identity function is one–one.
8. Into functions are not useful.
False. Into functions are widely used in applications.
9. Transcendental functions are not important for exams.
False. They are very important in calculus.
10. Classification of functions is only theoretical.
False. It plays a practical role in solving problems.
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