Functions and Their Graphs

Relations and Functions

Understanding the concepts of relations and functions is foundational in precalculus. A relation is a connection between input (x) and output (y) values, often represented as ordered pairs (x, y). A function is a special type of relation where each input has at most one output.

• Relation: Any set of ordered pairs (x, y).

• Function: Each x-value is paired with only one y-value.

• Vertical Line Test: A graph represents a function if no vertical line intersects the graph at more than one point.

• Example: The relation {(−3, 5), (0, 2), (3, 5)} is a function because each x-value is unique.

• Example: The relation {(2, 5), (0, 2), (2, 9)} is not a function because x = 2 is paired with two different y-values.

Verifying Functions from Equations

To determine if an equation represents a function, solve for y in terms of x. If each x yields only one y, it is a function. Function notation replaces y with f(x).

• Example: can be written as .

• If y has an even power (e.g., ), it is not a function.

• Linear equations (e.g., ) are functions.

Domain and Range of a Function

The domain of a function is the set of all possible input (x) values, and the range is the set of all possible output (y) values. To find the domain, project the graph onto the x-axis; for the range, project onto the y-axis.

• Interval Notation: Uses brackets [ ] for inclusive and parentheses ( ) for exclusive values.

• Set Builder Notation: Describes the domain and range using inequalities.

Finding Domain and Range from Equations

When given an equation, identify restrictions:

• For square roots: The expression inside must be non-negative.

• For fractions: The denominator must not be zero.

• Example: has domain .

• Example: has domain .

Transformations of Functions

Transformations change the appearance of a function's graph. The main types are reflections, shifts, and stretches/shrinks.

• Reflection: Flips the graph over the x-axis or y-axis. reflects over the x-axis.

• Shift: Moves the graph horizontally or vertically. shifts right by h and up by k.

• Stretch/Shrink: Multiplies the function by a constant c. stretches if , shrinks if .

Function Operations

Functions can be added, subtracted, multiplied, or divided. The domain of the resulting function is the intersection of the domains of the original functions, with additional restrictions for division (denominator ≠ 0).

• Add/Subtract: ,

• Multiply:

• Divide: ,

Function Composition and Decomposition

Function composition involves plugging one function into another: . The domain is restricted by both functions. Decomposition is expressing a function as a composition of two simpler functions.

• Example: If and , then .

• Decomposition: can be written as where and .

Additional info: These notes cover the essential concepts of Chapter 1 in Precalculus, including definitions, examples, and visual aids for understanding functions and their graphs, domain and range, and transformations.