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.
Compare relations that are functions with relations that are not 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 \(4\).
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.
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.
A function is an input-output machine: every input produces exactly one output.
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 \(y \in Y\).
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: 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: isolated points, written as a set of individual numbers. Continuous: connected line or curve, written as an inequality interval.
Absolute value function
The graph of \(f(x)=|x|\) forms a V-shape with vertex at \((0,0)\).
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 \(f(x)=-c\).
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.
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 \(|-3| = -(-3) = 3\).
Objective: Identify functions, vertical line test results, domain, codomain, range and key terms.
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?
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 ______.
Drag each term to the correct meaning. On a phone, tap a term first, then tap the matching box.
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