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
How the elements of the domain connect to the codomain.
One-one function (Injective)
Every element of the domain maps to a distinct element of the codomain. No two inputs share an output.
$f(x) = 2x + 1$ is one-one. $f(1) = 3$, $f(2) = 5$ — different inputs, different outputs.
Many-one function
Two or more elements of the domain map to the same element of the codomain.
$f(x) = x^2$ is many-one. $f(2) = 4$ and $f(-2) = 4$ — two different inputs, same output.
Onto function (Surjective)
Every element of the codomain is hit by at least one input. The range equals the codomain.
$f: \mathbb{R} \to \mathbb{R}$, $f(x) = x^3$ is onto over the reals — every real number is the cube of some real number.
Into function
At least one element of the codomain is not hit. The range is strictly inside the codomain.
$f: \mathbb{R} \to \mathbb{R}$, $f(x) = x^2$ is an into function over the reals — negative numbers in the codomain are never hit because $x^2 \geq 0$ always.
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. Bijections are exactly the functions that have a true inverse.
$f(x) = x + 5$ over the reals is a bijection.
Types of Functions Based on Degree
Polynomial functions, sorted by their highest power.
Type | Form | Example | Graph |
|---|---|---|---|
Constant function | $f(x) = c$ | $f(x) = 7$ | Horizontal line |
Identity function | $f(x) = x$ | $f(x) = x$ | Line through origin, slope 1 |
Linear function | $f(x) = ax + b$ | $f(x) = 2x + 3$ | Straight line, slope $a$ |
Quadratic function | $f(x) = ax^2 + bx + c$ | $f(x) = x^2 - 4$ | Parabola |
Cubic function | $f(x) = ax^3 + bx^2 + cx + d$ | $f(x) = x^3 - x$ | S-shaped curve |
Polynomial function | $f(x) = a_n x^n + \dots + a_1 x + a_0$ | $f(x) = x^4 - 3x + 1$ | Smooth curve with up to $n - 1$ turning points |
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
The big concept families.
Algebraic function
A function that can be written using a finite combination of addition, subtraction, multiplication, division, powers, and roots. Polynomials, rational functions, and root functions are all algebraic.
Example: $f(x) = \dfrac{x^2 + 1}{x - 3}$.
Trigonometric function
Functions defined from the geometry of the unit circle — $\sin x$, $\cos x$, $\tan x$, and their reciprocals $\csc x$, $\sec x$, $\cot x$. Periodic by nature.
Inverse trigonometric function
The inverses of the trigonometric functions: $\sin^{-1} x$, $\cos^{-1} x$, $\tan^{-1} x$. They undo the trig functions, with restricted domains to keep them one-one.
Exponential function
$f(x) = a^x$ for a positive constant $a$ — most commonly $a = e \approx 2.71828$.
Logarithmic function
$f(x) = \log_a x$ — the inverse of the exponential function. The natural log $\ln x$ uses base $e$.
Other Named Function Types
Several function types do not fit neatly into mapping, degree, or concept — they are named for the shape of their rule.
Type | Definition | Example |
|---|---|---|
Modulus (absolute value) function | $f(x) = |x|$ — output is always the non-negative version of the input | $f(-3) = 3$ |
Rational function | A ratio of polynomials: $f(x) = \dfrac{P(x)}{Q(x)}$ | $f(x) = \dfrac{1}{x}$ |
Signum function | Outputs $+1$, $0$, or $-1$ depending on sign of input | $\text{sgn}(-5) = -1$ |
Even function | $f(-x) = f(x)$ — graph is symmetric about the y-axis | $f(x) = x^2$ |
Odd function | $f(-x) = -f(x)$ — graph has rotational symmetry about the origin | $f(x) = x^3$ |
Periodic function | Repeats: $f(x + T) = f(x)$ for some period $T$ | $f(x) = \sin x$ has period $2\pi$ |
Greatest integer (floor) function | Outputs the largest integer ≤ input | $\lfloor 3.7 \rfloor = 3$ |
Inverse function | Undoes the original function: if $f(a) = b$, then $f^{-1}(b) = a$ | $f(x) = x + 5$ has $f^{-1}(x) = x - 5$ |
Composite function | Apply one function to the result of another: $(f \circ g)(x) = f(g(x))$ | If $f(x) = x^2$ and $g(x) = x + 1$, then $(f \circ g)(x) = (x + 1)^2$ |
How Do You Classify a Function? Three Worked Examples
We walk through three problems — Quick, Standard, and Stretch.
Quick example
Quick. Classify $f(x) = 5x - 2$.
Degree-based: linear (highest power of $x$ is $1$).
Mapping-based: one-one (different inputs give different outputs).
Concept-based: algebraic (a polynomial).
Final answer: linear, one-one, algebraic.
The detour students take
Standard. Is $f(x) = x^2$ a one-one function on $\mathbb{R}$?
Wrong path. A rusher looks at the formula, sees a single output for each input, and writes:
"Every $x$ gives one $f(x)$, so the function is one-one."
That confuses "function" with "one-one function." The first means each input has one output; the second means each input has a different output.
Correct path. Test for one-one: pick two different inputs and check whether their outputs match.
$$f(2) = 4, \quad f(-2) = 4$$
Two different inputs, the same output. So $f(x) = x^2$ is not one-one on $\mathbb{R}$ — it is many-one.
It becomes one-one if you restrict the domain to $\mathbb{R}_{\geq 0}$.
In Bhanzu's Grade 11 cohorts, the "every $x$ gives one $f(x)$ so it's one-one" slip shows up on roughly three out of ten first attempts at function classification. A Bhanzu trainer who hears this draws a horizontal line through $y = 4$ on the parabola — two intersection points — and the test lands instantly.
Stretch example
Stretch. Is $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = 2x - 3$ a bijection?
One-one check. If $f(a) = f(b)$, then $2a - 3 = 2b - 3 \Rightarrow a = b$. Different inputs give different outputs. One-one ✓.
Onto check. For any $y$ in the codomain $\mathbb{R}$, can we find an $x$ with $f(x) = y$? Solve $2x - 3 = y \Rightarrow x = \dfrac{y + 3}{2}$. The $x$ exists for every real $y$. Onto ✓.
Both conditions met.
Final answer: $f(x) = 2x - 3$ is a bijection on $\mathbb{R}$, with inverse $f^{-1}(x) = \dfrac{x + 3}{2}$.
Why Do Types of Functions Matter?
Function classification is not a tagging exercise. It tells you what you can do with the function.
Inverses. Only bijections have true inverses. To invert a many-one function, you must restrict its domain first.
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 — line, parabola, S-curve, periodic wave — before plotting a single point.
Calculus. The derivative of $x^n$ follows a power rule; the derivative of $\sin x$ is $\cos x$; the derivative of $e^x$ is $e^x$. Classification tells you which rule to apply.
Real-world modelling. Population growth uses exponential functions. Sound waves and light intensity use trigonometric functions. Cost-revenue analysis uses quadratics. The classification picks the model.
The Mistakes Students Make Most Often
Three errors account for most of the marks lost on function-classification questions.
Mistake 1: Confusing "function" with "one-one function."
Where it slips in: Every "function" maps each input to one output. That does not make it one-one — one-one requires different outputs for different inputs.
Don't do this: Concluding $f(x) = x^2$ is one-one because every $x$ has one $f(x)$.
The correct way: Test for one-one by checking if any two different inputs share an output. $f(2) = f(-2) = 4$, so $x^2$ is not one-one on $\mathbb{R}$.
Mistake 2: Forgetting the codomain matters for onto.
Where it slips in: Students check whether the range looks reasonable without comparing it to the declared codomain.
Don't do this: Declaring $f: \mathbb{R} \to \mathbb{R}{\geq 0}$, $f(x) = x^2$ as not-onto because "$x^2$ misses negative numbers." (But the codomain here is $\mathbb{R}{\geq 0}$, which excludes negatives.)
The correct way: Always compare the range to the declared codomain. The same rule can be onto or not depending on how the codomain is specified.
Mistake 3: Confusing even functions with odd-numbered exponents.
Where it slips in: "Even function" sounds like it should mean "polynomial of even degree."
Don't do this: Concluding $f(x) = x^2 + 1$ is odd because it has a constant term.
The correct way: Even function means $f(-x) = f(x)$; odd function means $f(-x) = -f(x)$. $x^2 + 1$ is even because $(-x)^2 + 1 = x^2 + 1$. The constant term does not matter — the symmetry test does.
The Mathematician Who Shaped The Function Concept
Leonhard Euler (1707–1783, Switzerland). Euler introduced the modern notation $f(x)$ for a function in his 1734 paper, and his 1748 textbook Introductio in Analysin Infinitorum developed the systematic theory of functions that calculus rests on. The classification of functions by degree, by concept, and by inverse-existence — all three start with Euler's framework.
Conclusion
Functions are classified along three axes: mapping, degree, and concept — the same function can have multiple classifications.
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, and the named extras (modulus, rational, periodic, greatest-integer, composite).
Classification is not a tagging exercise — it tells you what tools (inverses, derivatives, real-world models) you can apply to the function.
A Practical Next Step
Three problems to practise. If you stall, come back to the classification table above.
Classify $f(x) = x^3 + 2x$ by degree, mapping (on $\mathbb{R}$), and concept.
Determine whether $f(x) = 3x + 1$ on $\mathbb{R}$ is a bijection, and if so, find its inverse.
State whether $f(x) = x^4 - x^2$ is even, odd, or neither, and justify with the symmetry test.
Want a Bhanzu trainer to walk through more classification problems live? Book a free demo class — online globally.
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