Functions and Graphs

Relations, Functions, Domain, and Range

Understanding the distinction between relations and functions is foundational in precalculus. A relation is any set of ordered pairs, while a function is a relation in which each input (domain value) corresponds to exactly one output (range value).

• Domain: The set of all possible input values (x-values) for a function.

• Range: The set of all possible output values (y-values) for a function.

• Set Notation: Domains and ranges are often expressed in set notation, e.g., {x | x ≥ 0}.

To determine if a relation is a function, check that no input value is paired with more than one output value.

Increasing, Decreasing, and Constant Intervals

The behavior of a function can be described by intervals where it is increasing, decreasing, or constant:

• Increasing: A function f(x) is increasing on an interval if, as x increases, f(x) also increases.

• Decreasing: A function f(x) is decreasing on an interval if, as x increases, f(x) decreases.

• Constant: A function f(x) is constant on an interval if, as x increases, f(x) remains the same.

Even and Odd Functions

Functions can be classified as even, odd, or neither based on their symmetry:

• Even Function: f(x) is even if for all x in the domain. Even functions are symmetric about the y-axis.

• Odd Function: f(x) is odd if for all x in the domain. Odd functions have rotational symmetry about the origin.

• Neither: If a function does not satisfy either condition, it is neither even nor odd.

Piecewise Functions

A piecewise function is defined by different expressions over different intervals of the domain. To evaluate a piecewise function, determine which interval the input belongs to and use the corresponding expression.

• Example:

Relative Maximum and Minimum

Relative (or local) maxima and minima are important for understanding the behavior of functions:

• Relative Maximum: A function value f(a) is a relative maximum if it is greater than all nearby values.

• Relative Minimum: A function value f(b) is a relative minimum if it is less than all nearby values.

Graphical Analysis of Functions

Graphs provide visual insight into the properties of functions, including intercepts, intervals of increase/decrease, and extrema.

• x-intercept: The point(s) where the graph crosses the x-axis (set y = 0).

• y-intercept: The point where the graph crosses the y-axis (set x = 0).

• Relative Maxima/Minima: Highest/lowest points in a local region of the graph.

Linear Functions and Slope

Slope and Rate of Change

The slope of a line measures its steepness and is calculated as:

• Parallel Lines: Have the same slope.

• Perpendicular Lines: Have slopes that are negative reciprocals; their product is -1.

Forms of Linear Equations

• Standard Form:

• Slope-Intercept Form:

• Point-Slope Form:

Special cases include horizontal lines (, slope = 0) and vertical lines (, slope undefined).

Finding Intercepts

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

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

Average Rate of Change

The average rate of change of a function between two points is the slope of the secant line connecting those points:

Difference Quotient

Definition and Application

The difference quotient is a formula used to compute the average rate of change over an interval and is foundational for calculus:

, for

• Used to find the slope of the secant line between two points on a function.

Transformations of Functions

Reflections

• Reflection about the x-axis: reflects the graph over the x-axis.

• Reflection about the y-axis: reflects the graph over the y-axis.

Vertical and Horizontal Shifts

• Vertical Shift Up: shifts the graph up by c units.

• Vertical Shift Down: shifts the graph down by c units.

• Horizontal Shift Left: shifts the graph left by c units.

• Horizontal Shift Right: shifts the graph right by c units.

Vertical Stretching and Shrinking

• Vertical Stretch: , stretches the graph vertically by a factor of c.

• Vertical Shrink: , shrinks the graph vertically by a factor of c.

Horizontal Stretching and Shrinking

• Horizontal Shrink: , shrinks the graph horizontally by a factor of c.

• Horizontal Stretch: , stretches the graph horizontally by a factor of c.

Summary Table of Transformations

The following table summarizes the main types of function transformations and their effects:

Practice and Application

Apply these concepts to analyze, graph, and transform functions. Practice problems may involve identifying intervals of increase/decrease, finding intercepts, classifying functions as even/odd/neither, and describing or sketching transformations.