Types of Functions: Complete Guide with Examples and Graphs

Last updated: 01/20/2026

Introduction

Functions are all around us, from checking phone battery life to calculating costs while shopping. In math, a function assigns each input exactly one output, written as y = f(x). A key property is uniqueness: every input has one output. The vertical line test helps verify functions on a graph. Functions can be classified by mapping, degree, mathematical concepts, or special properties. This guide will explain each type of function to help students understand and apply them confidently to real-world problems.

Types of Functions Based on Mapping

One-to-One Function (Injective)

A one-to-one function assigns different outputs to different inputs. In other words, if f(a) = f(b), then a = b.

Definition: Each input maps to a unique output.
Example: f(x) = 2x + 1 Here, no two values of x produce the same y.
Key Property: Passes the horizontal line test.
Domain and Range: Both are sets of real numbers.
Graph Description: A straight line that never repeats a y-value.

Real-life connection: Temperature conversion from Celsius to Fahrenheit is one-to-one.

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Many-to-One Function

In a many-to-one function, different inputs can produce the same output.

Definition: Multiple inputs map to a single output.
Example: f(x) = x² f(2) = 4 and f(−2) = 4.
Key Property: Fails the horizontal line test.
Domain: All real numbers.
Range: Non-negative real numbers.
Graph Description: A U-shaped parabola symmetric about the y-axis.

Onto Function (Surjective)

An onto function covers every possible value in the codomain.

Definition: Every element in the codomain has at least one pre-image.
Example: f(x) = x³, from ℝ to ℝ.
Property: Range equals codomain.
Graph Description: A smooth curve extending infinitely in both directions.

Into Function

An into function does not cover all elements of the codomain.

Definition: Range is a proper subset of the codomain.
Example: f(x) = x², from ℝ to ℝ.
Graph Description: Parabola that never goes below the x-axis.

Types of Functions Based on Degree

Identity Function

Definition: Output equals input.
Formula: f(x) = x
Graph Description: A straight line passing through the origin with slope 1.
Property: Domain = Range = ℝ.

Constant Function

Definition: Output is always the same.
Formula: f(x) = c
Example: f(x) = 5
Graph Description: Horizontal line.
Property: Domain is ℝ; range is a single value.

Linear Function

Definition: Degree 1 polynomial.
Formula: f(x) = mx + b
Example: f(x) = 2x + 3
Graph Description: Straight line with slope m.
Application: Cost calculations, speed-distance relations.

Quadratic Function

Definition: Degree 2 polynomial.
Formula: f(x) = ax² + bx + c
Example: f(x) = x² − 4x + 3
Graph Description: Parabola opening upward or downward.
Property: Has a maximum or minimum point (vertex).

Cubic Function

Definition: Degree 3 polynomial.
Formula: f(x) = ax³ + bx² + cx + d
Example: f(x) = x³
Graph Description: S-shaped curve.
Application: Modeling volume and growth trends.

Polynomial Function

Definition: Sum of powers of x with real coefficients.
Example: f(x) = 3x⁴ − 2x² + x − 7
Graph Description: Smooth curve without breaks.
Property: Domain is all real numbers.

Types of Functions Based on Mathematical Concepts

Algebraic Functions

Functions formed using algebraic operations.

Example: f(x) = √(x + 2)
Domain: x ≥ −2
Graph Description: Curve starting at x = −2 and increasing.

Trigonometric Functions

Based on angles and triangles.

Common Types: sin x, cos x, tan x
Example: f(x) = sin x
Graph Description: Wave-like curve.
Property: Periodic behavior.

Inverse Trigonometric Functions

Reverse of trigonometric functions.

Example: f(x) = sin⁻¹(x)
Domain: −1 ≤ x ≤ 1
Graph Description: Smooth increasing curve.

Exponential Functions

Definition: Variable appears in the exponent.
Formula: f(x) = aˣ, a > 0
Example: f(x) = 2ˣ
Graph Description: Rapid growth curve.
Application: Population growth, compound interest.

Logarithmic Functions

Definition: Inverse of exponential functions.
Formula: f(x) = logₐ x
Example: f(x) = log₁₀ x
Graph Description: Slow growth curve.
Domain: x > 0

Miscellaneous Types of Functions

Modulus Function

Formula: f(x) = |x|
Graph Description: V-shaped graph.
Property: Always non-negative.

Rational Function

Definition: Ratio of two polynomials.
Example: f(x) = 1 / (x − 1)
Graph Description: Curve with asymptotes.
Domain: Excludes values that make denominator zero.

Signum Function

Formula: f(x) = −1 (x < 0), 0 (x = 0), 1 (x > 0)
Graph Description: Step-like graph.

Even and Odd Functions

Even: f(−x) = f(x), symmetric about y-axis (e.g., x²).
Odd: f(−x) = −f(x), symmetric about origin (e.g., x³).

Periodic Function

Definition: Repeats values at regular intervals.
Example: sin x
Graph Description: Repeating wave.

Greatest Integer Function

Formula: f(x) = ⌊x⌋
Graph Description: Step function.

Inverse Function

Definition: Reverses input and output.
Property: Only exists for one-to-one functions.
Graph Description: Reflection across y = x.

Composite Function

Definition: One function inside another.
Formula: (f ∘ g)(x) = f(g(x))
Example: f(x) = x², g(x) = x + 1 → f(g(x)) = (x + 1)²

Summary Table of Function Types

Category	Function Type	Definition	Example	Domain	Range
Based on Mapping	One-to-One (Injective)	Each input maps to unique output	f(x) = 2x + 5	ℝ	ℝ
	Many-to-One	Multiple inputs map to same output	f(x) = x²	ℝ	[0, ∞)
	Onto (Surjective)	Every codomain element has pre-image	f(x) = x + 3	ℝ	ℝ
	Bijective	One-to-one and onto	f(x) = 3x – 2	ℝ	ℝ
	Into	Some codomain elements have no pre-image	f(x) = x² + 1	ℝ	[1, ∞)
Based on Degree	Identity	Output = Input	f(x) = x	ℝ	ℝ
	Constant	All inputs map to constant	f(x) = 4	ℝ	{4}
	Linear	Degree 1 polynomial	f(x) = 2x – 7	ℝ	ℝ
	Quadratic	Degree 2 polynomial	f(x) = 3x² + 2x – 1	ℝ	[-4/3, ∞)
	Cubic	Degree 3 polynomial	f(x) = 2x³ – 5x² + x + 3	ℝ	ℝ
Based on Concepts	Trigonometric	Angle-based ratios	f(θ) = sinθ	ℝ	[-1, 1]
	Logarithmic	Inverse of exponential	f(x) = log₂x	(0, ∞)	ℝ
	Exponential	Variable in exponent	f(x) = 2ˣ	ℝ	(0, ∞)
Miscellaneous	Modulus	Absolute value of input	f(x) = \|x\|	ℝ	[0, ∞)
	Rational	Ratio of two polynomials	f(x) = (x+2)/(x-3)	ℝ \ {3}	ℝ \ {1}
	Even	f(-x) = f(x)	f(x) = x²	ℝ	[0, ∞)
	Odd	f(-x) = -f(x)	f(x) = x³	ℝ	ℝ
	Composite	Combination of two functions	f(g(x)) = 2x² + 3	ℝ	[3, ∞)

Conclusion

Functions are the language of relationships—whether you’re calculating distance, analyzing data, or solving real-world problems. From simple linear functions to complex trigonometric waves, each type has unique properties that make it useful for specific scenarios. By understanding how functions are classified (by mapping, degree, math concepts, or miscellaneous traits), you’ll be able to identify, graph, and solve problems involving any function with confidence.

Remember, practice is key! Try graphing different functions, finding their domains and ranges, or computing composite and inverse functions to reinforce your skills. Functions aren’t just a math topic—they’re a tool to make sense of the world around you. Keep exploring, and you’ll master them in no time!

FAQs About Types of Functions

1. How to identify a one-to-one function?

Use the horizontal line test: If any horizontal line intersects the graph more than once, it’s not one-to-one. Algebraically, if f(x1)=f(x2) implies x1=x2, the function is one-to-one.

2. What is the difference between even and odd functions?

Even functions satisfy f(−x)=f(x) and are symmetric about the y-axis (e.g., x2).
Odd functions satisfy f(−x)=−f(x) and are symmetric about the origin (e.g., x3).

3. Do all functions have inverses?

No—only bijective functions (one-to-one and onto) have inverses. If a function is not one-to-one (e.g., x2), it doesn’t have a unique inverse unless its domain is restricted.

4. What is the vertical line test used for?

The vertical line test confirms if a relation is a function. If any vertical line intersects the graph more than once, the relation has multiple outputs for one input—so it’s not a function.

5. How do you find the domain of a rational function?

The domain of a rational function f(x)=Q(x)P(x) is all real numbers except where the denominator Q(x)=0 (since division by zero is undefined). For example, f(x)=x−21 has a domain of R∖{2}.

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James | WuKong Math Teacher

Graduated from Columbia University in the United States and has rich practical experience in mathematics competitions’ teaching, including Math Kangaroo, AMC… He teaches students the ways to flexible thinking and quick thinking in sloving math questions, and he is good at inspiring and guiding students to think about mathematical problems and find solutions.