1.1 Functions
In our daily lives, many quantities depend on one or more variables, such as total wages depending on days worked, or total price depending on the mass of items purchased. In mathematics, a function is a special type of relation between two sets where every input element is linked to exactly one output element. Understanding functions provides the foundation for representing relationships mathematically through equations, graphs, and mappings.
Learning Objectives
- Explain the meaning of a function and represent functions using arrow diagrams, graphs, and algebraic notations such as \(f: x \to y\) or \(f(x) = y\).
- Determine whether a relation is a function using arrow diagrams and the vertical line test on graphs.
- Determine the domain, codomain, and range of discrete and continuous functions, including absolute value functions.
- Calculate the image of a function when an object is given, and determine the object(s) when an image is given.
Classification of Functions: Arrow Diagrams and the Vertical Line Test
Compare relations that are functions with relations that are not functions.
Functions
One-to-one: each object has one distinct image.
Many-to-one: two or more objects share the same image.
Example: \(f(x)=x^2\) maps both \(-2\) and \(2\) to \(4\).
Not Functions
One-to-many: an object has more than one image.
Many-to-many: multiple objects share multiple images.
Example: a circle fails the vertical line test because one \(x\)-value maps to two \(y\)-values.
| Relation type | Arrow diagram behaviour | Function? |
|---|---|---|
| One-to-one | Every object has exactly one distinct image. | Yes |
| Many-to-one | Two or more objects share the same image. | Yes |
| One-to-many | One object has more than one image. | No |
| Many-to-many | Multiple objects share multiple images. | No |
| Missing image | An element in the domain has no arrow at all. | No |
Vertical Line Test
Draw a vertical line anywhere across the graph.
Count the number of intersection points with the curve.
If it cuts at only one point, the relation is a function.
If it cuts at more than one point, the relation is not a function.
Short Explanation
A function is an input-output machine: every input produces exactly one output.
Definition of a Function
A function relating set \(X\) to set \(Y\) is a special relation where each element \(x \in X\) is mapped to one and only one element \(y \in Y\).
Functional Notation
Arrow notation: \(f: x \to 2x + 1\). Equation notation: \(f(x) = 2x + 1\). Here, \(x\) is the object and \(f(x)\) or \(y\) is the image.
Domain, Codomain and Range
Domain: set of all possible input values \(x\). Codomain: set of all possible target values \(y\). Range: set of actual images obtained from the domain. Range \(\subseteq\) Codomain.
Discrete vs Continuous
Discrete: isolated points, written as a set of individual numbers. Continuous: connected line or curve, written as an inequality interval.
Absolute value function
\(|x| = \begin{cases} x & \text{if } x \ge 0 \\ -x & \text{if } x < 0 \end{cases}\)The graph of \(f(x)=|x|\) forms a V-shape with vertex at \((0,0)\).
Finding Images and Objects
To find an image, substitute the given \(x\) into \(f(x)\).
To find an object, set \(f(x)=k\) and solve for \(x\).
Mapping onto itself occurs when \(f(x)=x\).
For \(|f(x)|=c\), solve \(f(x)=c\) or \(f(x)=-c\).
Try to Answer First
Answer in your mind, then press "Check Answer".
Determine whether the relation represented by the ordered pairs \(\{(2, 4), (3, 9), (4, 16), (2, 8)\}\) is a function. Explain your answer.
Given the function \(f(x) = |2x - 5|\), find: (a) the image of \(x = 1\), (b) the values of \(x\) such that \(f(x) = 7\).
Based on an arrow diagram, set \(X = \{1, 2, 3\}\) is mapped to set \(Y = \{2, 4, 6, 8\}\) where \(1 \to 2\), \(2 \to 4\), and \(3 \to 6\). State the domain, codomain, and range of this function.
Common Mistakes
- Confusing codomain with range: writing all elements of set \(Y\) as the range, even though some elements in \(Y\) do not have an object mapped to them. Range only includes elements that are actual images.
- Forgetting the negative root in absolute value equations: when solving \(|ax + b| = c\), students often only solve \(ax + b = c\) and omit \(ax + b = -c\).
- Misinterpreting the vertical line test: believing a relation is not a function if an asymptote exists where the vertical line touches no point. A function only fails the test if a vertical line cuts the graph at more than one point.
- Writing continuous intervals as discrete sets: writing the domain or range of a continuous line graph as a list of integers \(\{1, 2, 3, 4\}\) instead of an inequality interval \(1 \le x \le 4\).
Concept Misunderstandings
A function cannot have two different objects sharing the same image.
A many-to-one relation is a valid function, for example \(f(x)=x^2\) where \(f(-2)=4\) and \(f(2)=4\). It is only a one-to-many relation that is not a function.
Every element in the codomain must be connected to an object.
The codomain can contain unused elements. Only the elements in the domain are required to have an image.
\(|x|\) is always equal to \(x\).
The modulus \(|x|\) yields the positive numerical value of \(x\). If \(x\) is negative, \(|x| = -x\), which turns the negative number into a positive value, for example \(|-3| = -(-3) = 3\).
Summary
- A function maps every element in set \(X\) to one and only one element in set \(Y\).
- Valid function relations: one-to-one and many-to-one.
- Non-function relations: one-to-many and many-to-many.
- Vertical Line Test: A relation graph is a function if and only if every vertical line cuts the graph at at most one point.
- Domain: set of all input objects \(x\). Codomain: set of all potential output elements \(y\). Range: set of actual images mapped from the domain, with Range \(\subseteq\) Codomain.
- For absolute value \(|x|\), output is always non-negative, \(|x| \ge 0\).
Short Activity
Objective: Identify functions, vertical line test results, domain, codomain, range and key terms.
A. Objective Quiz
1 Which of the following relations represents a function?
2 A vertical line test is conducted on the graph of a circle defined by \(x^2 + y^2 = 9\). What is the result and conclusion of the test?
B. Fill in the Blanks
3 In function notation \(f(x) = y\), the input element \(x\) is called the object, while the resulting output value \(y\) is called the ______.
4 The set of all actual obtained output values from substituting every element of the domain into a function is called the ______.
C. Matching / Drag and Drop
Drag each term to the correct meaning. On a phone, tap a term first, then tap the matching box.
