Relations and functions are key concepts in algebra, connecting 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 in various fields.

Relations and Functions

Domain and range of relations

• Domain: all possible input values (x-values) for a relation
  • For a relation in a table, the domain is all x-values listed (time, age)
  • For a relation represented by a graph, the domain is all x-values that have a corresponding y-value (temperature, distance)
  • For a relation defined by an equation, the domain is all real numbers that result in real y-values when substituted for x (prices, weights)
• Range: all possible output values (y-values) for a relation
  • For a relation in a table, the range is all y-values listed (cost, height)
  • For a relation represented by a graph, the range is all y-values that have a corresponding x-value (speed, volume)
  • For a relation defined by an equation, the range is all real numbers that result when each x-value from the domain is substituted into the equation (area, profit)

Relations vs functions

• Relation: set of ordered pairs (x, y) that defines a relationship between two quantities
  • Can be represented by a table, graph, or equation (mapping, association)
  • In a relation, each x-value can be paired with multiple y-values (student grades, car prices)
• Function: special type of relation where each x-value (input) is paired with exactly one y-value (output)
  • Vertical line test: if a vertical line intersects a graph more than once, the relation is not a function (parabola, circle)
  • In a function, each x-value in the domain is paired with a unique y-value in the range (linear equation, exponential growth)
  • Functions can be represented using function notation: $f(x)$, where $f$ is the function name and $x$ is the input variable (quadratic function, sine function)
  • A function that is both one-to-one and onto is called a one-to-one correspondence

Function evaluation for inputs

• To evaluate a function for a specific input value, substitute the given value for the input variable (usually $x$) in the function's equation
  1. Example: Given $f(x) = 2x + 3$, find $f(5)$
     • Substitute 5 for x: $f(5) = 2(5) + 3$
     • Simplify: $f(5) = 10 + 3 = 13$
  2. Example: Given $g(x) = x^2 - 4$, find $g(-2)$
     • Substitute -2 for x: $g(-2) = (-2)^2 - 4$
     • Simplify: $g(-2) = 4 - 4 = 0$
• The result of the evaluation is the output value (y-value) corresponding to the given input value (coordinate pair, point on graph)
• When given a function in a table or graph form, find the output value that corresponds to the given input value
  • For a table, locate the row with the given input value and find the corresponding output value (spreadsheet, data set)
  • For a graph, locate the point on the graph with the given x-value and determine the corresponding y-value (coordinate plane, curve)

Variables in Functions

• Independent variable: the input variable in a function, typically represented by x
• Dependent variable: the output variable in a function, typically represented by y, whose value depends on the independent variable