Relations and Functions

Definitions and Examples

Understanding the concepts of relations and functions is fundamental in precalculus. A relation is any set of ordered pairs (x, y), while a function is a special type of relation where each input x has exactly one output y.

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

• Function: Each x corresponds to exactly one y.

• Example of a Function: {(1, 4), (2, 5), (3, 6)} — each x has one output.

• Non-Example: {(1, 4), (1, 7), (2, 5)} — x = 1 has two outputs; not a function.

Vertical Line Test

The vertical line test is a graphical method to determine if a relation is a function. If every vertical line intersects the graph at most once, the graph represents a function.

• Two or more intersections with a vertical line indicate the graph is not a function.

Function Evaluation

Substitution and Examples

To evaluate a function, substitute the given value for every x in the function's expression.

• To find , replace every x with a.

• For , replace every x with .

• Example: If , then:

• Tip: Use parentheses around negative values to avoid errors.

Domain and Range

Definitions and Interval Notation

The domain of a function is the set of all possible x-values, and the range is the set of all corresponding y-values. Interval notation is used to describe these sets.

• Interval Notation:

  • (a, b):

  • [a, b]:

  • (a, b]:

  • [a, b):

  • (-\infty, a):

  • [a, \infty):

• Open circle: endpoint is not included.

• Closed circle: endpoint is included.

Domain Rules

• Denominator: Cannot be zero.

• Even-root: Radicand must be greater than or equal to zero.

Example 1: Requires so Domain:

Example 2: Requires so Domain:

Intercepts

x-intercept and y-intercept

Intercepts are points where the graph crosses the axes.

• x-intercept: — set y = 0 and solve for x.

• y-intercept: — set x = 0 and solve for y.

• A graph can have zero, one, or multiple x-intercepts, but at most one y-intercept.

Example: Find x-intercept: Set Find y-intercept: Set So, x-intercept: , y-intercept:

Even, Odd, and Neither Functions

Definitions and Symmetry

Functions can be classified based on their symmetry properties.

• Even Function: — symmetric about the y-axis.

• Odd Function: — symmetric about the origin.

• Neither: If neither condition is met.

Example (Even):

Example (Odd):

Example (Neither):

Increasing, Decreasing, and Constant Functions

Behavior of Functions

The behavior of a function can be described as increasing, decreasing, or constant over intervals.

• Increasing: Graph goes up as x increases.

• Decreasing: Graph goes down as x increases.

• Constant: Graph remains horizontal as x increases.

Example: Decreasing on , increasing on

Minima and Maxima

Local and Absolute Extrema

Functions can have local and absolute maximum and minimum values.

• Local Maximum: A nearby peak.

• Local Minimum: A nearby valley.

• Absolute Maximum: Highest y-value on the domain.

• Absolute Minimum: Lowest y-value on the domain.

Example: Local and absolute minimum at ; no local or absolute maximum.

Basic Function Shapes

Common Functions

Recognizing basic function shapes is important for graphing and analysis.

• Linear:

• Quadratic:

• Absolute Value:

• Square Root:

Function Transformations

Types of Transformations

Functions can be transformed by shifting, stretching, compressing, or reflecting their graphs.

Example: The graph of , shifted 3 units left, reflected across the x-axis, and shifted 5 units up: Start with $y = \sqrt{x}$ Shift left: Reflect x-axis: Shift up:

Practice Problems

• Find the domain of

• Check for symmetry:

• Graph a function that is increasing on , has a local maximum at , an absolute maximum at , and exactly 3 x-intercepts.

• Find x and y intercepts of

• Write the equation for the graph of shifted 3 units left, reflected across the x-axis, and shifted 5 units up.