Functions and Graphs

Increasing, Decreasing, and Constant Functions

Understanding how a function behaves on different intervals is fundamental in analyzing its graph. A function can be classified as increasing, decreasing, or constant on specific intervals of its domain.

• Increasing Function: A function f is increasing on an open interval I if for any two numbers x_1 and x_2 in I, with x_1 < x_2, we have f(x_1) < f(x_2).

• Decreasing Function: A function f is decreasing on an open interval I if for any x_1 < x_2 in I, f(x_1) > f(x_2).

• Constant Function: A function f is constant on an open interval I if for any x_1, x_2 in I, f(x_1) = f(x_2).

Example: Consider a function whose graph rises from left to right on the interval (-∞, 0), is flat on (0, 2), and falls on (2, ∞). It is increasing on (-∞, 0), constant on (0, 2), and decreasing on (2, ∞).

Relative Maxima and Minima

Relative (or local) maxima and minima are points where a function reaches a highest or lowest value within a certain interval.

• Relative Maximum: A function value f(a) is a relative maximum if there exists an open interval containing a such that f(a) > f(x) for all x in that interval.

• Relative Minimum: A function value f(b) is a relative minimum if there exists an open interval containing b such that f(b) < f(x) for all x in that interval.

Example: On the graph of a function, a peak represents a relative maximum, and a valley represents a relative minimum.

Tests for Symmetry

Symmetry helps in sketching and understanding the behavior of graphs. The three main types of symmetry are with respect to the y-axis, x-axis, and the origin.

• Y-axis Symmetry: Replace x with -x in the equation. If the equation remains unchanged, the graph is symmetric about the y-axis.

• X-axis Symmetry: Replace y with -y. If the equation remains unchanged, the graph is symmetric about the x-axis.

• Origin Symmetry: Replace x with -x and y with -y. If the equation remains unchanged, the graph is symmetric about the origin.

Example: The graph of y = x^2 is symmetric about the y-axis, while y = x^3 is symmetric about the origin.

Even and Odd Functions

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

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

• Odd Function: f is odd if f(-x) = -f(x) for all x in the domain. The graph is symmetric about the origin.

• Neither: If neither condition holds, the function is neither even nor odd.

Example: f(x) = x^2 is even; f(x) = x^3 is odd; f(x) = x^2 + x is neither.

Piecewise Functions

A piecewise function is defined by different expressions over different parts of its domain. These functions are useful for modeling situations where a rule changes based on the input value.

• Each "piece" of the function applies to a specific interval of the domain.

• To evaluate a piecewise function at a given value, determine which interval the value falls into and use the corresponding expression.

Example: If f(x) = \begin{cases} x^2, & x < 0 \\ 2x + 1, & x \geq 0 \end{cases}, then f(-2) = 4 and f(1) = 3.

Difference Quotient

The difference quotient is a fundamental concept in calculus, used to compute the average rate of change of a function over an interval. It is defined as:

• It measures the change in the function's value as the input changes by h.

• Simplifying the difference quotient is a common algebraic exercise.

Example: If f(x) = x^2, then