Relations and Functions Overview

Relations

• A relation is a set of ordered pairs.
• It can be represented as points on a coordinate plane, a table, a mapping diagram, or a graph.
• Relations can be any set of pairs. Functions have specific restrictions.
• Every relation has a domain (x-values) and a range (y-values). The domain is input, the range is output.
• A relation can have multiple outputs for a single input.
• A relation can be expressed in set notation, like {(1, 2), (2, 4), (3, 6)}.
• Arrow notation graphically represents a relation, with arrows connecting inputs to outputs.

Functions

• A function is a specific type of relation.
• Each input (x-value) is paired with exactly one output (y-value).
• Functions can be represented as sets of ordered pairs (e.g., {(1, 2), (3, 4), (5, 6)}).
• Tables display input-output pairings.
• Graphs can be curves or sets of points.
• Rules or equations (e.g., y = 2x + 1) represent functions.
• The vertical line test determines if a graph is a function. If a vertical line intersects the graph more than once, it's not a function.
• The domain of a function is the set of all possible input values (x-values).
• The range of a function is the set of all possible output values (y-values).
• Functions can be described with notation like f(x) = 2x + 1.
• In y = f(x), x is the independent variable and y is the dependent variable.

Key Differences Between Relations and Functions

• A function's key attribute is that each input corresponds to exactly one output. A relation allows for multiple outputs per input.
• For a relation to be a function, each x-value can have only one y-value.
• The vertical line test visually confirms if a graph is a function.

Types of Functions

• Linear functions: Straight-line graphs; general form y = mx + b (m = slope, b = y-intercept).
• Quadratic functions: Parabola graphs; general form y = ax² + bx + c (a, b, c are constants).
• Polynomial functions: Involve variables raised to non-negative integer powers.
• Rational functions: Quotients of two polynomial functions.
• Exponential functions: Involve an exponent where the base is constant and the exponent is a variable.
• Logarithmic functions: Inverses of exponential functions.

Evaluating Functions

• Evaluating a function means substituting a specific input value (x) to find the corresponding output (y or f(x)).
• Substitute the given value for x and calculate. For instance, if f(x) = 2x + 1 and you want f(3), substitute 3 for x to get f(3) = 7.

Domain and Range of Functions

• The domain of a function is the set of all possible input values (x-values).
• The range of a function is the set of all possible output values (y-values).
• When finding the domain and range, consider if any input values result in undefined outputs (e.g., division by zero or the square root of a negative number).