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Relations and Functions

MAT1501 - Fundamental Mathematics · MODULE 3: GRAPHS

Relations and Functions

In this topic, you will learn about relations and functions, which are fundamental concepts in mathematics. Understanding these concepts is essential for solving problems in algebra and calculus. You will learn how to identify relations and functions, determine their domains and ranges, and apply these ideas in various mathematical contexts.

Key idea: After studying this topic, you should be able to:

  • Identify the type of correspondence in any relation.
  • Identify the domain and range in a given relation.
  • Recognise when a relation is a function.
  • Determine function values by substituting given domain values into the formula that defines the function.
  • Identify the natural domain of a function.
  • Identify dependent and independent variables in a specific example, and use functional notation correctly.
  • Recognise when two functions are equal.

Understanding Relations

A relation is a set of ordered pairs. In mathematics, we often represent these pairs as (x, y), where x is the first element (input) and y is the second element (output). For example, the relation {(1, 2), (2, 3), (3, 4)} consists of three ordered pairs.

Example 1: Identifying a Relation

Consider the following set of points: {(0, 1), (1, 2), (2, 3)}. This set is a relation because it consists of ordered pairs.

Domain and Range

The domain of a relation is the set of all first coordinates (x-values), while the range is the set of all second coordinates (y-values).

Example 2: Finding Domain and Range

For the relation {(1, 2), (2, 3), (3, 4)}, the domain is {1, 2, 3} and the range is {2, 3, 4}.

Tip: When identifying the domain and range, make sure to list each value only once, even if it appears multiple times in the relation.

Functions

A function is a special type of relation where each input (x-value) is associated with exactly one output (y-value). This means that for every x in the domain, there is one and only one corresponding y in the range.

Example 3: Identifying a Function

Consider the relation {(1, 2), (2, 3), (3, 4)}. This is a function because each x-value corresponds to one y-value. However, the relation {(1, 2), (1, 3)} is not a function because the input 1 corresponds to two different outputs (2 and 3).

Functional Notation

We use functional notation to express the relationship between the input and output. If f is a function, we write f(x) to denote the output corresponding to the input x. For example, if f(x) = 2x + 1, then f(2) = 2(2) + 1 = 5.

Example 4: Using Functional Notation

If f(x) = x^2, then f(3) = 3^2 = 9.

Natural Domain of a Function

The natural domain of a function is the set of all input values for which the function is defined. For example, if a function involves a square root, the input must be non-negative because the square root of a negative number is not a real number.

Example 5: Finding the Natural Domain

For the function g(x) = √(x - 1), the natural domain is x ≥ 1 because the expression inside the square root must be non-negative.

Watch out: Always check for values that may make the function undefined, such as division by zero or square roots of negative numbers.

Identifying Dependent and Independent Variables

In a function, the independent variable is the input (usually x), while the dependent variable is the output (usually y). The value of the dependent variable depends on the value of the independent variable.

Example 6: Identifying Variables

In the function f(x) = 3x + 2, x is the independent variable, and f(x) is the dependent variable.

Recognising Equal Functions

Two functions are considered equal if they have the same domain and produce the same output for every input in that domain. This means that if f(x) = g(x) for all x in their common domain, then f and g are equal functions.

Example 7: Equal Functions

Let f(x) = x^2 and g(x) = (x)(x). These two functions are equal because they produce the same output for every x.

Summary

  • A relation is a set of ordered pairs.
  • The domain is the set of all first coordinates, and the range is the set of all second coordinates.
  • A function is a relation where each input corresponds to exactly one output.
  • Functional notation expresses the relationship between input and output.
  • The natural domain includes all input values for which the function is defined.
  • Dependent variables depend on independent variables.
  • Two functions are equal if they have the same domain and produce the same output for all inputs.

Check your understanding

  1. What is the difference between a relation and a function?
  2. How do you determine the domain and range of a relation?
  3. What is the natural domain of the function h(x) = 1/(x - 2)?
  4. Give an example of two functions that are equal.