Types of Functions: Classification, Graphs and Examples

What Are The Types of Functions?

A function $f: A \to B$ is a rule that assigns every element of the set $A$ exactly one element of the set $B$, and the types of functions describe how that rule behaves. They are classified in four main ways: by how inputs are paired with outputs (mapping), by the highest power in the rule (degree), by the kind of output (range), and by the family the rule belongs to (algebraic, trigonometric, exponential, logarithmic). A few more names describe symmetry (even, odd, periodic) or how functions combine (inverse, composite).

Three words carry every classification below:

• Domain: the set $A$ of allowed inputs.

• Codomain: the set $B$ the outputs are declared to lie in.

• Range: the outputs that actually occur. The range is always inside the codomain, but it can be smaller.

A function's type can depend on its domain and codomain, so every mapping example on this page states both. The notation $f(x)$ itself is Euler's: he introduced it in 1734 (MacTutor: Leonhard Euler). For the difference between a function and a general relation, see relations and functions.

One function can belong to several types at once: $f: \mathbb{R} \to \mathbb{R}$, $f(x) = x^3$ is cubic, bijective, algebraic and odd.

Types of Functions Based on Mapping

The two questions are independent, so every function on finite or infinite sets falls into one cell of this grid:

A constant function such as $f: \mathbb{R} \to \mathbb{R}$, $f(x) = 7$ is many-one and into: every input gives $7$, and no other number is ever an output.

Testing one-one and onto from a graph

The horizontal line test checks a graph of $f: \mathbb{R} \to \mathbb{R}$: draw horizontal lines $y = c$ across it. If some line crosses the graph more than once, $f$ is not one-one. If some line $y = c$, with $c$ in the codomain, misses the graph completely, $f$ is not onto. (The vertical line test is different: it checks whether a graph is a function at all.)

Types of Functions Based on Degree

These are polynomial functions, sorted by the highest power of $x$ with a non-zero coefficient.

The conditions matter. If $a = 0$ in $ax + b$, the rule is the constant $b$, not a linear function; if $a = 0$ in $ax^2 + bx + c$, the rule is at most linear. A polynomial of degree $n$ has at most $n$ real zeros. More on each type: polynomials, quadratic equations and cubic polynomials.

Types of Functions Based on Range

These functions are named for the kind of output they produce.

All three are many-one on $\mathbb{R}$: for example $\lvert 3 \rvert = \lvert -3 \rvert$ and $\lfloor 3.2 \rfloor = \lfloor 3.9 \rfloor$.

Types of Functions Based on Family

The base conditions are not decoration. With $a = 1$, $1^x = 1$ for every $x$, which is a constant function; with $a < 0$, a value like $(-4)^{\frac{1}{2}}$ is not a real number.

Other Types: Even, Odd, Periodic, Inverse and Composite

What The Type Tells You

• Inverses: only a bijective function has an inverse. To invert $x^2$, first restrict it to $f: [0, \infty) \to [0, \infty)$; its inverse is then $\sqrt{x}$.

• Graph shape: the type predicts the graph before you plot a point (see the grid above).

• Number of solutions: a linear equation $ax + b = 0$ with $a \neq 0$ has exactly one solution; a polynomial equation of degree $n$ has at most $n$ real solutions.

• Modelling: periodic functions model tides, sound waves and the length of daylight across a year; exponential functions model compound interest and early, unchecked population growth.

Examples of Types of Functions

Example 1

Show that $f: \mathbb{R} \to \mathbb{R}$, $f(x) = 2x - 3$ is bijective, and find its inverse.

One-one: suppose $f(a) = f(b)$.

$$2a - 3 = 2b - 3$$

$$2a = 2b$$

$$a = b$$

Onto: take any $y$ in the codomain $\mathbb{R}$ and solve $f(x) = y$.

$$2x - 3 = y$$

$$x = \frac{y + 3}{2}$$

This $x$ is a real number for every real $y$, so every element of the codomain is an output.

Final answer: $f$ is bijective (and linear, since $a = 2 \neq 0$), with $f^{-1}(x) = \frac{x + 3}{2}$.

Example 2

Is $f: \mathbb{R} \to \mathbb{R}$, $f(x) = x^2$ one-one?

The tempting argument: "every $x$ gives exactly one value of $x^2$, so the function is one-one." That sentence describes every function. One-one asks something stronger: do different inputs always give different outputs?

Test two different inputs:

$$f(2) = 4$$

$$f(-2) = 4$$

Two inputs share the output $4$, so $f$ is many-one. It is also into, because no real input gives $-1$.

Final answer: not one-one. Restricted to $f: [0, \infty) \to [0, \infty)$, the same rule becomes bijective.

Example 3

Is $g(x) = x^3 - x$ even, odd or neither? What about $h(x) = x^2 + x$?

For $g$, replace $x$ with $-x$:

$$g(-x) = (-x)^3 - (-x)$$

$$g(-x) = -x^3 + x$$

$$g(-x) = -(x^3 - x) = -g(x)$$

So $g$ is odd. For $h$, test $x = 1$:

$$h(1) = 1 + 1 = 2$$

$$h(-1) = 1 - 1 = 0$$

Since $h(-1) \neq h(1)$ and $h(-1) \neq -h(1)$, $h$ is neither even nor odd.

Final answer: $g$ is odd; $h$ is neither.

Example 4

Let $f(x) = x^2$ and $g(x) = x + 1$. Find $(f \circ g)(3)$ and $(g \circ f)(3)$.

$$(f \circ g)(3) = f(g(3)) = f(4) = 16$$

$$(g \circ f)(3) = g(f(3)) = g(9) = 10$$

Final answer: $16$ and $10$. The order of a composite changes the result.

Example 5

Find $\lfloor -2.3 \rfloor$, $\lfloor 4 \rfloor$ and $\operatorname{sgn}(-0.5)$.

The first instinct on $\lfloor -2.3 \rfloor$ is to drop the decimal and write $-2$. But $-2$ is greater than $-2.3$, and the floor must be less than or equal to the input. On the number line, the integers either side of $-2.3$ are $-3$ and $-2$, and only $-3$ is at or below it.

$$\lfloor -2.3 \rfloor = -3$$

$$\lfloor 4 \rfloor = 4$$

$$\operatorname{sgn}(-0.5) = -1$$

Final answer: $-3$, $4$ and $-1$.

Common mistakes with types of functions

Mistake 1: Deciding "onto" without looking at the codomain

Where it slips in: Questions that state the codomain in the notation $f: A \to B$, where students check the formula and skip the $B$.

Don't do this: Calling $f(x) = x^2$ "onto" or "not onto" from the formula alone.

The correct way: Compare the range with the stated codomain. For $f: \mathbb{R} \to \mathbb{R}$, $f(x) = x^2$ is not onto, because negative numbers have no pre-image (no input gives $-1$). For $f: \mathbb{R} \to [0, \infty)$, the same rule is onto, because every $y \geq 0$ equals $(\sqrt{y})^2$.

Mistake 2: Assuming even degree means even function

Where it slips in: Symmetry questions on polynomials, because "even" appears in both names.

Don't do this: Calling $f(x) = x^2 + x$ even because its degree is 2.

The correct way: Test $f(-x)$. Here $f(-x) = x^2 - x$, which is neither $f(x)$ nor $-f(x)$, so the function is neither even nor odd. A polynomial is even only when every power of $x$ in it is even (a constant counts as $x^0$).

Mistake 3: Ignoring the condition on the leading coefficient or base

Where it slips in: Classifying a rule written with letters, such as $f(x) = ax^2 + 3x + 1$ or $f(x) = a^x$, before checking the value of $a$.

Don't do this: Calling $f(x) = ax^2 + 3x + 1$ quadratic when $a = 0$, or calling $f(x) = 1^x$ exponential.

The correct way: Check the condition first. With $a = 0$, $ax^2 + 3x + 1 = 3x + 1$, which is linear. An exponential function needs $a > 0$ and $a \neq 1$, and $1^x = 1$ is constant.

Frequently Asked Questions

What are the main types of functions?

The types of functions are grouped by mapping (one-one, many-one, onto, into, bijective), by degree (constant, identity, linear, quadratic, cubic, polynomial), by range (modulus, signum, greatest integer) and by family (algebraic, trigonometric, inverse trigonometric, exponential, logarithmic). Even, odd, periodic, inverse and composite functions describe further properties.

What is the difference between an into function and an onto function?

An onto function reaches every element of its codomain. An into function misses at least one. Every function is exactly one of the two.

Can a function be both one-one and onto?

Yes. That is a bijective function, for example $f: \mathbb{R} \to \mathbb{R}$, $f(x) = 2x - 3$.

Is every linear function one-one?

Yes, as a function from $\mathbb{R}$ to $\mathbb{R}$. When $a \neq 0$, $f(x) = ax + b$ is one-one and onto, with inverse $f^{-1}(x) = \frac{x - b}{a}$. If $a = 0$, the rule is a constant function, which is many-one.

The Short Version

• The types of functions are grouped by mapping, degree, range and family, and one function can belong to several types.

• Mapping types (one-one, onto, bijective) depend on the stated domain and codomain, so always state both.

• Degree types need a non-zero leading coefficient; exponential and logarithmic functions need a base $a > 0$ with $a \neq 1$.

• Only bijective functions have inverses.

• Even and odd are symmetry tests on $f(-x)$, not statements about the degree.

Practice Problems With Answers

1. Classify $f: \mathbb{R} \to \mathbb{R}$, $f(x) = x^3 + 2x$ by degree, mapping, family and symmetry.

2. Show that $f: \mathbb{R} \to \mathbb{R}$, $f(x) = 3x + 1$ is bijective and find its inverse.

3. Is $f(x) = x^4 - x^2$ even, odd or neither?

4. Find $\lfloor -0.5 \rfloor$, $\operatorname{sgn}(-0.5)$ and $\lvert -0.5 \rvert$.

5. With $f(x) = x - 1$ and $g(x) = 2x$, find $(f \circ g)(4)$ and $(g \circ f)(4)$.

Answers:

1. Cubic (degree 3), bijective, algebraic and odd. One-one: if $a < b$ then $a^3 < b^3$ and $2a < 2b$, so $f(a) < f(b)$. Onto: a cubic on $\mathbb{R}$ takes every real value. Odd: $f(-x) = -x^3 - 2x = -f(x)$.

2. From $3a + 1 = 3b + 1$ we get $a = b$; solving $3x + 1 = y$ gives $x = \frac{y - 1}{3}$ for every real $y$. So $f$ is bijective with $f^{-1}(x) = \frac{x - 1}{3}$.

3. Even: $f(-x) = (-x)^4 - (-x)^2 = x^4 - x^2 = f(x)$.

4. $\lfloor -0.5 \rfloor = -1$, $\operatorname{sgn}(-0.5) = -1$ and $\lvert -0.5 \rvert = 0.5$.

5. $(f \circ g)(4) = f(8) = 7$ and $(g \circ f)(4) = g(3) = 6$.

If Example 2 or Mistake 1 still feels slippery, redo them with the mapping diagrams above. Want a live Bhanzu trainer to work through more function problems? Book a free demo class.