Functions and Relations: Definitions, Properties, and Notation

Section 2.1: Functions and Relations

Definition of a Relationship

A relationship in mathematics is any set of ordered pairs. The domain is the set of all first components (inputs), and the range is the set of all second components (outputs) of the relationship.

• Domain: The set of all possible input values (first elements of ordered pairs).

• Range: The set of all possible output values (second elements of ordered pairs).

• Example: For the set {(0, 9.1), (10, 6.7), (20, 10.7), (30, 13.2), (42, 21.7)}, Domain = {0, 10, 20, 30, 42} Range = {9.1, 6.7, 10.7, 13.2, 21.7}

• Explanatory Variables: Another term for domain (often called independent variables).

• Response Variables: Another term for range (often called dependent variables).

Definition of a Function

A function is a special type of relationship where each element in the domain corresponds to exactly one element in the range.

• Formal Definition: A function is a correspondence from a first set (domain) to a second set (range) such that each element in the domain is paired with exactly one element in the range.

• Example of Not a Function: Set A = {(1, 2), (3, 4), (5, 6), (5, 8)} Here, 5 in the domain is paired with both 6 and 8 in the range, violating the definition of a function.

• Example of a Function: Set B = {(1, 2), (3, 4), (6, 5), (8, 5)} Each domain value is paired with only one range value, even if multiple domain values share the same range value.

Determining Whether an Equation Represents a Function

To determine if an equation represents a function, solve for the dependent variable and check if each input yields only one output.

• Independent Variable: Also called explanatory variable or input (commonly 'x').

• Dependent Variable: Also called response variable or output (commonly 'y').

• Example 1: $x^2 + y = 4$ Solve for y: $y = -x^2 + 4$ For each x, there is only one y. This is a function.

• Example 2: $x^2 + y^2 = 4$ Solve for y: $y^2 = 4 - x^2$ $y = \pm \sqrt{4 - x^2}$ For some x, there are two possible y values. This is not a function.

Function Notation

Function notation is a concise way to represent functions. The notation f(x) denotes the output of the function f for input x.

• Standard Notation: $y = f(x)$

• Example: $f(x) = 0.012x^2 - 0.2x + 8.7$ To evaluate $f(30)$: $f(30) = 0.012 \times 30^2 - 0.2 \times 30 + 8.7$ $= 0.012 \times 900 - 6 + 8.7$ $= 10.8 - 6 + 8.7$ $= 13.5$

Evaluating Functions

To evaluate a function, substitute the input value into the function's formula.

• Example: $f(x) = x^2 - 3x + 5$ $f(2) = 2^2 - 3 \times 2 + 5 = 4 - 6 + 5 = 3$

• Evaluating at Expressions: $f(x+3)$ Substitute $x+3$ for $x$: $f(x+3) = (x+3)^2 - 3(x+3) + 5$ $= x^2 + 6x + 9 - 3x - 9 + 5$ $= x^2 + 3x + 5$

• Evaluating at Negative Inputs: $f(-x)$ Substitute $-x$ for $x$: $f(-x) = (-x)^2 - 3(-x) + 5$ $= x^2 + 3x + 5$

Graphing Functions

Functions can be represented graphically. A table of values helps plot points for the function.

• Example: $f(x) = 2x$ and $f(x) = 2x + 4$ Table of Values:

x	f(x) = 2x	f(x) = 2x + 4
-2	-4	0
-1	-2	2
0	0	4
1	2	6
2	4	8

Vertical Line Test

The vertical line test is a graphical method to determine if a curve represents a function. If any vertical line intersects the graph at more than one point, the graph does not represent a function.

• Function: Each vertical line crosses the graph at most once.

• Not a Function: A vertical line crosses the graph more than once.

Interval Notation and Set Builder Notation

These notations are used to describe the domain and range of functions.

• Interval Notation: Uses brackets and parentheses to indicate the range of values. Examples:

  • Including endpoints: $[-4, 6]$

  • Excluding endpoints: $(-2, 4]$

• Set Builder Notation: Describes sets using a condition. Example: $\{x \mid -4 \leq x \leq 6\}$ means all x such that $-4 \leq x \leq 6$.

Examples of Domain and Range

• Example 1: Domain: $x \mid -4 \leq x \leq 6$ (Interval: $[-4, 6]$) Range: $y \mid 1 \leq y \leq 3$ (Interval: $[1, 3]$)

• Example 2: Domain: $x \mid -2 < x \leq 4$ (Interval: $(-2, 4]$) Range: $y \mid -4 < y \leq 4$ (Interval: $(-4, 4]$)

Additional info: The notes cover foundational concepts in functions and relations, including definitions, notation, evaluation, and graphical representation, which are essential for Precalculus students.