Function: Notations and Rules

What is a Function?

A function is a special kind of relation where every input gives you exactly one output. If you pick any element from the domain and run it through the function, you always land on a single, definite value. There is no ambiguity, no branching into two different results.

Formally, \( f \) is a function if and only if two conditions hold:

1. Every member of \( f \) is an ordered pair. That is, \( xfy \Rightarrow (x, y) \in f \).
2. If both \( (x, y) \in f \) and \( (x, z) \in f \), then \( y = z \). No input maps to two different outputs.

Here is a quick way to think about it. Consider the set \( f = \{(1, 3), (2, 5), (3, 7)\} \). Each first coordinate (1, 2, 3) appears exactly once, so this is a valid function. But if you had \( g = \{(1, 3), (1, 5), (2, 7)\} \), the input 1 maps to both 3 and 5. That violates the second condition, so \( g \) is not a function.

Notation for Functions

When you write \( (x, y) \in f \), you are saying that the ordered pair \( (x, y) \) belongs to the function \( f \). Here, \( x \) is the argument (or input) and \( y \) is the image (or output). The older notation \( xfy \) expresses the same relationship, though you will rarely see it in modern textbooks.

The set of all valid arguments is called the domain of the function, and the set of all images (the actual outputs you get) is called the range. Keep these two terms sharp in your mind because they come up constantly.

Several equivalent notations exist for expressing \( (x, y) \in f \):

• \( y = f(x) \) is the standard modern notation you will use most often.
• \( y = fx \) is a shorthand sometimes used in algebra and analysis.
• \( y : xf \) and \( y = x^f \) are older notational forms you may encounter in classical texts.

For example, if \( f(x) = 2x + 1 \), then \( f(3) = 7 \). The argument is 3, the image is 7, and you write \( (3, 7) \in f \).

Domain and Range

The domain of a function \( f \) is the set of all inputs for which \( f \) is defined. The range (also called the image) is the set of all outputs that \( f \) actually produces. These are not the same as the codomain, which is the larger set that the range sits inside.

Consider \( f(x) = x^2 \) where \( x \in \mathbb{R} \). The domain is all of \( \mathbb{R} \), but the range is only \( [0, \infty) \) because squaring a real number never gives a negative result. If you are told that \( f : \mathbb{R} \to \mathbb{R} \), the codomain is \( \mathbb{R} \), but the range is still just the non-negative reals.

Understanding this distinction is critical. When you encounter a function, your first question should always be: what is its domain, and what values does it actually take?

Into Function

A function \( f \) is said to be into \( Y \) if the range of \( f \) is a subset of \( Y \), but does not necessarily cover all of \( Y \). In set notation, \( R_f \subset Y \). Some elements of \( Y \) are left “unhit” by the function.

For example, let \( f : \{1, 2, 3\} \to \{a, b, c, d\} \) be defined by \( f(1) = a \), \( f(2) = b \), \( f(3) = c \). The range is \( \{a, b, c\} \), which is a proper subset of the codomain \( \{a, b, c, d\} \). The element \( d \) has no preimage, so \( f \) is an into function. You can think of it as a function that maps “into” the codomain without filling it completely.

Onto Function

A function \( f \) is onto \( Y \) (also called surjective) if the range of \( f \) equals \( Y \) exactly. In set notation, \( R_f = Y \). Every element in the codomain is hit by at least one element from the domain.

For example, let \( g : \{1, 2, 3, 4\} \to \{a, b, c\} \) be defined by \( g(1) = a \), \( g(2) = b \), \( g(3) = c \), \( g(4) = a \). The range is \( \{a, b, c\} \), which equals the codomain. Every element in the codomain has at least one preimage, so \( g \) is onto. Notice that \( a \) is hit twice, that is perfectly fine for a surjection.

The general notation for a mapping is \( f : X \to Y \), where \( X \) is the domain and \( Y \) is the codomain. Whether the function is into or onto depends on whether the range fills up all of \( Y \).

One-to-One Function

A function is one-to-one (also called injective) if it maps distinct elements to distinct elements. No two different inputs ever produce the same output. Formally, \( f \) is one-to-one if and only if \( x_1 \ne x_2 \Rightarrow f(x_1) \ne f(x_2) \). Equivalently, \( f(x_1) = f(x_2) \Rightarrow x_1 = x_2 \).

Consider \( f(x) = 3x + 2 \) defined on \( \mathbb{R} \). If \( f(x_1) = f(x_2) \), then \( 3x_1 + 2 = 3x_2 + 2 \), which gives \( x_1 = x_2 \). So \( f \) is one-to-one. Now consider \( g(x) = x^2 \) on \( \mathbb{R} \). Here \( g(2) = 4 \) and \( g(-2) = 4 \), so two different inputs give the same output. This function is not one-to-one.

A function that is both one-to-one and onto is called a bijection. Bijections are especially important because they have inverses: you can uniquely reverse the mapping from output back to input.

Restriction of a Function

If you have a function \( f : X \to Y \) and you want to limit its domain to a smaller set \( A \subseteq X \), you get the restriction of \( f \) to \( A \). Formally, the restriction is \( f \cap (A \times Y) \), which is usually written as \( f|_A \). It is still a function from \( A \) into \( Y \), but it only “sees” the inputs in \( A \).

For example, let \( f : \mathbb{R} \to \mathbb{R} \) be defined by \( f(x) = x^2 \). If you restrict \( f \) to \( A = [0, \infty) \), you get \( f|_{[0,\infty)}(x) = x^2 \) for \( x \ge 0 \). This restricted version is actually one-to-one (since you have removed the negative inputs that caused duplicates), and it is a common technique for making a function invertible.

Extension of a Function

An extension goes in the opposite direction. A function \( f \) is an extension of a function \( g \) if \( g \subseteq f \), meaning that \( f \) agrees with \( g \) everywhere that \( g \) is defined, but \( f \) may also be defined on additional inputs outside the domain of \( g \).

For example, let \( g : \{1, 2\} \to \mathbb{R} \) with \( g(1) = 5 \) and \( g(2) = 8 \). Now define \( f : \{1, 2, 3\} \to \mathbb{R} \) with \( f(1) = 5 \), \( f(2) = 8 \), and \( f(3) = 11 \). Since \( f \) matches \( g \) on inputs 1 and 2 and adds a new mapping for 3, we say \( f \) is an extension of \( g \). Conversely, \( g \) is the restriction of \( f \) to \( \{1, 2\} \). These two concepts are mirror images of each other.

Operations on Functions

Once you have two functions defined on the same domain, you can combine them using arithmetic operations. If \( f \) and \( g \) are both functions from \( X \) to \( \mathbb{R} \), you can define the following:

• Addition: \( (f + g)(x) = f(x) + g(x) \)
• Subtraction: \( (f – g)(x) = f(x) – g(x) \)
• Multiplication: \( (f \cdot g)(x) = f(x) \cdot g(x) \)
• Division: \( \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} \), provided \( g(x) \ne 0 \)

The domain of the combined function is the intersection of the domains of \( f \) and \( g \). For division, you must also exclude any \( x \) where \( g(x) = 0 \).

As a concrete example, let \( f(x) = x + 1 \) and \( g(x) = x – 2 \), both defined on \( \mathbb{R} \). Then \( (f + g)(x) = 2x – 1 \), \( (f \cdot g)(x) = (x+1)(x-2) = x^2 – x – 2 \), and \( \left(\frac{f}{g}\right)(x) = \frac{x+1}{x-2} \) for all \( x \ne 2 \).

Key Takeaways

Here are the main ideas you should carry away from this page:

• A function maps each input to exactly one output. The uniqueness of the output is what distinguishes a function from a general relation.
• The standard notation \( y = f(x) \) tells you that \( x \) is the input and \( y \) is the output. The domain is the set of valid inputs, and the range is the set of actual outputs.
• An into function does not cover its entire codomain. An onto (surjective) function does.
• A one-to-one (injective) function never sends two different inputs to the same output. A function that is both one-to-one and onto is a bijection.
• Restriction shrinks the domain. Extension expands it. These are inverse operations and are commonly used to make functions invertible or to define piecewise behavior.
• You can add, subtract, multiply, and divide functions, as long as you respect the domains. For division, always exclude points where the denominator is zero.

Frequently Asked Questions