Functions and Graphs

Definitions and Properties of Functions

Understanding the behavior and characteristics of functions is fundamental in precalculus. The following concepts are essential for analyzing functions on intervals and their graphical representations.

• Increasing Function: A function f is increasing on an interval if for any two points x and y in the interval, with x < y, then f(x) < f(y).

• Decreasing Function: A function f is decreasing on an interval if for any two points x and y in the interval, with x < y, then f(x) > f(y).

• Constant Function: A function f is constant on an interval if for any two points x and y in the interval, f(x) = f(y).

Example: Consider f(x) = x^2 on the interval [0, 2]. Since f(x) increases as x increases, f is increasing on [0, 2].

Relative Extrema

Relative extrema refer to the highest or lowest points in a specific interval of a function.

• Relative Maximum: A function f has a relative maximum at x = a if f(a) > f(x) for all x in an open interval containing a.

• Relative Minimum: A function f has a relative minimum at x = b if f(b) < f(x) for all x in an open interval containing b.

Example: For f(x) = x^2, the point (0, 0) is a relative minimum.

Symmetry of Functions

Types of Symmetry

Symmetry in functions helps classify their graphs and predict their behavior.

• Y-axis Symmetry (Even Functions): A function is symmetric about the y-axis if replacing x with -x yields the same function: for all x in the domain.

• Origin Symmetry (Odd Functions): A function is symmetric about the origin if replacing x with -x yields the negative of the function: for all x in the domain.

• X-axis Symmetry: A graph is symmetric about the x-axis if replacing y with -y yields the same equation. Note: Functions cannot be symmetric about the x-axis, as this would violate the definition of a function (each input has only one output).

Example: f(x) = x^2 is even, since .

Example: f(x) = x^3 is odd, since .

Testing for Symmetry

• Replace x with -x in the function and compare to the original.

• Replace y with -y in the equation and compare to the original.

• Replace both x and y with their negatives for origin symmetry.

Example: For y = x^2:

• Y-axis symmetry: Replace x with -x: (same as original).

• Origin symmetry: Replace x and y with -x and -y: , which is not the same as .

Summary Table: Symmetry Tests

Additional info:

• Some content was inferred from context and standard precalculus topics, such as the definitions of increasing/decreasing functions and symmetry tests.

• Relative extrema are also known as local maxima and minima.

• Functions cannot be symmetric about the x-axis, as this would violate the vertical line test.