Types of Functions in Maths – Graphs & Examples

Types of functions define the various ways mathematical inputs, known as the domain, are mapped to specific outputs, known as the codomain or range. In mathematics, a function is a rule where every element in set A corresponds to exactly one element in set B, categorized by their algebraic properties, graphical behaviors, and mapping characteristics.

Exploring the Different Types of Functions in Math

To understand types of functions in math, think of a function as a machine. You put something in (the input), the machine follows a specific rule, and it spits something out (the output). The most important rule is that for every single input you provide, the machine can only give you one specific output. If you press the "Coke" button on a vending machine and it sometimes gives you a Sprite, it isn't functioning correctly. In math, we use f(x) to describe this "machine" at work.

When we look at types of functions with examples, we generally divide them into two major categories. The first category looks at how the sets are "mapped"—basically, how the inputs and outputs are paired up. The second category looks at the algebraic expressions themselves, like linear, quadratic, or absolute value functions.

While the provided reference sources focus on mathematical theory, it is a common best-practice opinion to note that types of functions in python or other programming languages mirror these mathematical concepts. In coding, you define a function with specific parameters (inputs) and return values (outputs). Whether you are analyzing types of functions graphs in a textbook or writing a line of code, the underlying logic of mapping remains the same.

Classification Based on Set Mapping (Injective, Surjective, Bijective)

Mapping is all about the "loyalty" or relationship between the input set (Domain) and the output set (Codomain).

One-to-One Function (Injective Function)

A function is called one-to-one if every element in the domain maps to its own unique element in the codomain. This means no two different inputs can ever share the same output.

• Example: If f(x) = x + 5, every unique value of x will produce a unique result.

Onto Function (Surjective Function)

A function is "onto" if every single element in the codomain has at least one corresponding element in the domain. In other words, no element in the output set is left "lonely" or unused. The range is exactly equal to the codomain.

Many-to-One Function

In this type, two or more different elements from the domain can map to the same single element in the codomain.

• Example: f(x) = x squared. Both f(2) and f(-2) result in 4.

Bijective Function

A function that is both one-to-one (injective) and onto (surjective) is called a bijective function. This creates a perfect one-to-one correspondence where every input has exactly one output, and every output is paired with exactly one input.

Types of Functions Graphs and Algebraic Definitions

Graphs are the visual heartbeat of functions. They allow us to see the relationship between numbers as a shape or a curve on a coordinate plane.

1. Identity Function

The identity function is the "honest" function. Whatever you put in is exactly what comes out.

• Equation: f(x) = x

• Graph: A straight diagonal line passing through the origin (0,0) at a 45-degree angle. Every point on the graph has matching coordinates like (1,1), (2,2), or (-5,-5).

2. Constant Function

The constant function is the "stubborn" function. No matter what input you provide, the output never changes.

• Equation: f(x) = c (where c is a constant like 5 or 10)

• Graph: A perfectly horizontal line parallel to the x-axis.

3. Linear Function

A linear function is a polynomial of degree one. It is the most common type of function used in daily life to calculate things like hourly wages or travel distance.

• Equation: f(x) = mx + c

• Graph: A straight line where the slope (m) tells you how steep it is and the y-intercept (c) tells you where it crosses the vertical axis.

4. Quadratic Function

A quadratic function is a polynomial of degree two. Its graph is a distinct U-shaped curve.

• Equation: f(x) = ax squared + bx + c

• Graph: This curve is called a parabola. If "a" is positive, the U opens upward (like a smile). If "a" is negative, it opens downward (like a frown).

5. Polynomial Function

Polynomial functions involve non-negative integer powers of x. They are expressed as a sum of terms and can have multiple "turns" or curves depending on their degree.

• General Form: f(x) = anx to the power of n + ... + a1x + a0

6. Rational Function

A rational function is essentially a fraction where both the numerator and the denominator are polynomials.

• Equation: f(x) = P(x) / Q(x), where Q(x) is not zero.

• Graph: These graphs usually have "breaks" or gaps called asymptotes where the function cannot exist because the denominator would be zero.

7. Absolute Value Function

Also known as the modulus function, this function gives the non-negative distance of a number from zero. It essentially "strips away" negative signs.

• Equation: f(x) = |x|

• Graph: A sharp V-shaped curve with its vertex at the origin.

8. Greatest Integer Function (Step Function)

This function rounds an input down to the nearest integer that is less than or equal to the number.

• Equation: f(x) = [x]

• Graph: It looks like a series of disjointed horizontal steps, which is why it is often called a "step function."

Special Classifications: Even, Odd, and Periodic

Functions can be classified on the basis of their symmetrical properties and periodic nature.

• Even Functions:  A function is even when f(-x) equals f(x). A graph is the mirror image of an even function when flipped over the y-axis. A very common example of an even function is when f(x) equals x squared.

• Odd Functions: An odd function is a type of function that satisfies the equation: f(-x) = -f(x). Examples include any function that follows the equation f(x) = x cubed. Odd functions have rotational symmetry.

• Periodic Functions: The function takes on periodic values after certain intervals of time. The best example for periodic functions is trigonometric functions, which include sin(x) and cos(x). They hold a unique place in functions because of their periodic behavior.

Practical Takeaways: Why Does This Matter?

Learning these functions is more about mastering the "rules" of the world rather than simply passing some kind of math assessment.

• Linear functions: assist us with analyzing constant growth, for instance, a savings account that increases by a fixed amount each month.

• Quadratic function: describes the movement of a ball projected in space.

• Periodic functions: have been used to represent everything from sound and light waves to the tides of the ocean.

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