Types of Functions — Complete Classification with Examples
Types of Functions — Complete Classification with Examples
What is a Function?
A function is a rule that assigns to every input exactly one output. Written as f(x), where x is the input and f(x) is the output.
The set of allowed inputs is the domain. The set of resulting outputs is the range. The collection that the outputs sit inside (the "target") is the codomain.
A useful test for whether a rule is a function: every input maps to exactly one output. The vertical line test on a graph is the visual version of this rule.
How Types of Functions are Classified
Functions are sorted along three axes — and the same function can be classified in multiple ways at once.
| Axis | Question being asked | Types under it |
|---|---|---|
| Mapping | How do inputs connect to outputs? | One-one, many-one, onto, into, bijection |
| Degree | What is the highest power of x? | Constant, linear, quadratic, cubic, polynomial |
| Concept | What mathematical family does the rule belong to? | Algebraic, trigonometric, exponential, logarithmic, and more |
We will walk through each group below, with the type-completeness covering every named function a student meets through Grade 12.
Types of Functions Based on Mapping
One-one function (Injective)
Every element of the domain maps to a distinct element of the codomain. No two inputs share an output.
Example: f(x)=2x+1.
Many-one function
Two or more elements of the domain map to the same element of the codomain.
Example: f(x)=x2.
Onto function (Surjective)
Every element of the codomain is hit by at least one input. The range equals the codomain.
Example: f:R→R, f(x)=x3.
Into function
At least one element of the codomain is not hit. The range is strictly inside the codomain.
Example: f:R→R, f(x)=x2.
Bijection (One-one and onto)
A function that is both one-one and onto. Every input maps to a distinct output, and every output is achieved.
Example: f(x)=x+5.
Types of Functions Based on Degree
Polynomial functions, sorted by their highest power.
| Type | Form | Example |
|---|---|---|
| Constant function | f(x)=c | f(x)=7 |
| Identity function | f(x)=x | f(x)=x |
| Linear function | f(x)=ax+b | f(x)=2x+3 |
| Quadratic function | f(x)=ax2+bx+c | f(x)=x2−4 |
| Cubic function | f(x)=ax3+bx2+cx+d | f(x)=x3−x |
| Polynomial function | f(x)=anxn+⋯+a1x+a0 | f(x)=x4−3x+1 |
The constant function is the degree-zero case; the identity is a specific linear function with slope 1 and intercept 0.
Types of Functions Based on Math Concepts
Algebraic function
A function that can be written using a finite combination of addition, subtraction, multiplication, division, powers, and roots.
Example: f(x)=x2+1/(x−3).
Trigonometric function
Functions defined from the geometry of the unit circle — sin(x), cos(x), tan(x), and their reciprocals.
Inverse trigonometric function
The inverses of the trigonometric functions: sin−1(x), cos−1(x), tan−1(x).
Exponential function
Example: f(x)=a^x for a positive constant a.
Logarithmic function
Example: f(x)=log_a(x).
Other Named Function Types
| Type | Definition | Example |
|---|---|---|
| Modulus (absolute value) function | f(x)= | x |
| Rational function | A ratio of polynomials: f(x)=P(x)/Q(x) | f(x)=1/x |
| Signum function | Outputs +1, 0, or -1 depending on sign of input | sgn(−5)=−1 |
| Even function | f(−x)=f(x) — graph is symmetric about the y-axis | f(x)=x2 |
| Odd function | f(−x)=−f(x) — graph has rotational symmetry about the origin | f(x)=x3 |
| Periodic function | Repeats: f(x+T)=f(x) for some period T | f(x)=sin(x) |
| Greatest integer (floor) function | Outputs the largest integer ≤ input | ⌊3.7⌋=3 |
| Inverse function | Undoes the original function | f(x)=x+5 has f−1(x)=x−5 |
| Composite function | Apply one function to the result of another | If f(x)=x2 and g(x)=x+1, then (f∘g)(x)=(x+1)2 |
How Do You Classify a Function? Three Worked Examples
Quick example
Quick. Classify f(x)=5x−2.
- Degree-based: linear.
- Mapping-based: one-one.
- Concept-based: algebraic.
Final answer: linear, one-one, algebraic.
The detour students take
Standard. Is f(x)=x2 a one-one function on R?
Correct path. Test for one-one: pick two different inputs and check whether their outputs match.
Stretch example
Stretch. Is f:R→R defined by f(x)=2x−3 a bijection?
One-one check. If f(a)=f(b), then a=b. Different inputs give different outputs. One-one ✓.
Onto check. For any y in the codomain R, can we find an x with f(x)=y?
Both conditions met.
Final answer: f(x)=2x−3 is a bijection on R, with inverse f−1(x)=x+3/2.
Why Do Types of Functions Matter?
- Inverses. Only bijections have true inverses.
- Solving equations. Linear equations have one solution; quadratics have up to two; polynomial of degree n has up to n roots over the complex numbers.
- Graphing. Knowing the type tells you the shape before plotting a single point.
- Calculus. The derivative of x^n follows a power rule; the derivative of sin(x) is cos(x).
- Real-world modelling. Classification picks the model.
The Mistakes Students Make Most Often
- Confusing "function" with "one-one function."
- Forgetting the codomain matters for onto.
- Confusing even functions with odd-numbered exponents.
The Mathematician Who Shaped The Function Concept
Leonhard Euler (1707–1783, Switzerland).
Conclusion
- Functions are classified along three axes: mapping, degree, and concept.
- Mapping types track how inputs connect to outputs: one-one, many-one, onto, into, bijection.
- Degree types track the highest power: constant, linear, quadratic, cubic, polynomial.
- Concept families track the rule family: algebraic, trigonometric, exponential, logarithmic.
A Practical Next Step
- Classify f(x)=x3+2x by degree, mapping, and concept.
- Determine whether f(x)=3x+1 on R is a bijection, and if so, find its inverse.
- State whether f(x)=x4−x2 is even, odd, or neither, and justify with the symmetry test.