3.5 Relations and Functions

Relations and functions are key concepts in algebra that connect input values to output values. They're the building blocks for understanding how quantities relate to each other in mathematical models and real-world scenarios.

Domain and range define the possible inputs and outputs for relations and functions. Functions are special relations where each input has exactly one output. Understanding these ideas is crucial for analyzing and solving problems across many fields.

Relations and Functions

Domain and Range of Relations

A relation is any set of ordered pairs $(x, y)$. Every relation has a domain and a range.

Domain refers to all possible input values (x-values) for a relation.

• In a table, the domain is every x-value listed. For example, if a table shows hours worked and pay earned, the hours worked are the domain.
• On a graph, the domain is every x-value that has a corresponding point. You read it by looking at how far left and right the graph extends.
• In an equation, the domain is every real number you can substitute for $x$ and still get a real number output. For instance, in $y = \frac{1}{x}$, the domain is all real numbers except $x = 0$.

Range refers to all possible output values (y-values) for a relation.

• In a table, the range is every y-value listed.
• On a graph, the range is every y-value that appears. You read it by looking at how far up and down the graph extends.
• In an equation, the range is every real number that results when you plug domain values into the equation. For $y = x^2$, the range is $y \geq 0$ because squaring any number can never produce a negative result.

Relations vs. Functions

A relation is any set of ordered pairs $(x, y)$ that defines a relationship between two quantities. It can be represented by a table, graph, mapping diagram, or equation. In a relation, a single x-value can be paired with more than one y-value.

A function is a special type of relation where each x-value (input) is paired with exactly one y-value (output). This is the critical distinction: one input, one output. A single output can repeat for different inputs, but a single input can never produce two different outputs.

The Vertical Line Test is a quick way to check whether a graph represents a function:

1. Imagine drawing vertical lines across every part of the graph.
2. If any vertical line crosses the graph more than once, the relation is not a function.
3. If every vertical line crosses the graph at most once, it is a function.

For example, a parabola like $y = x^2$ passes the vertical line test (it's a function), but a circle like $x^2 + y^2 = 9$ fails it (not a function, because most x-values correspond to two y-values).

Functions can be written using function notation: $f(x)$, where $f$ names the function and $x$ is the input variable. This notation is useful because it clearly shows which input produces which output.

Function Evaluation for Inputs

To evaluate a function for a specific input, substitute the given value for $x$ in the function's equation, then simplify.

Example 1: Given $f(x) = 2x + 3$, find $f(5)$.

1. Substitute 5 for $x$: $f(5) = 2(5) + 3$
2. Simplify: $f(5) = 10 + 3 = 13$

So the point $(5, 13)$ lies on the graph of $f$.

Example 2: Given $g(x) = x^2 - 4$, find $g(-2)$.

1. Substitute $-2$ for $x$: $g(-2) = (-2)^2 - 4$

2. Simplify: $g(-2) = 4 - 4 = 0$

So the point $(-2, 0)$ lies on the graph of $g$.

Watch the signs carefully when substituting negative numbers. Always wrap the substituted value in parentheses to avoid sign errors, especially with exponents.

When a function is given as a table, locate the row with the given input and read off the corresponding output. When a function is given as a graph, find the input value on the x-axis, move vertically to the curve, and read the y-value at that point.

Variables in Functions

• Independent variable: the input variable, typically $x$. You choose its value freely.
• Dependent variable: the output variable, typically $y$ or $f(x)$. Its value depends on whatever you chose for the independent variable.

A helpful way to remember: the independent variable is what you control, and the dependent variable is what you measure or calculate as a result.