Functions and Their Domains

Understanding the Domain of a Function

The domain of a function is the set of all possible input values (usually x-values) for which the function is defined. When combining functions, such as F - G, the domain consists of all x-values that are in the domains of both F and G.

• Key Point: The domain of F - G is the intersection of the domains of F and G.

• Example: If F(x) is defined for x = -3, -2, -1, 0, 1, 2, 3 and G(x) is defined for x = -2, -1, 0, 1, 2, then the domain of F - G is x = -2, -1, 0, 1, 2.

Additional info: When functions are given as graphs with discrete points, the domain is the set of x-values where both functions have defined points.

Symmetry in Functions

Types of Symmetry

Symmetry in functions refers to how the graph of a function behaves with respect to the axes or the origin. Recognizing symmetry helps in graphing and understanding function properties.

• x-axis Symmetry: A graph has x-axis symmetry if reflecting it over the x-axis leaves it unchanged. For a function, this means replacing y with -y yields the same equation.

• y-axis Symmetry: A graph has y-axis symmetry if reflecting it over the y-axis leaves it unchanged. For a function, this means f(-x) = f(x) for all x in the domain (even function).

• Origin Symmetry: A graph has origin symmetry if rotating it 180 degrees about the origin leaves it unchanged. For a function, this means f(-x) = -f(x) for all x in the domain (odd function).

Example:

• The function f(x) = x2 has y-axis symmetry.

• The function f(x) = x3 has origin symmetry.

• The equation y2 = x has x-axis symmetry (not a function).

Sketching Symmetric Functions

How to Sketch Functions with Specific Symmetries

To sketch a function or shape with a particular symmetry, use the definitions above:

• x-axis symmetry: For every point (x, y), include (x, -y).

• y-axis symmetry: For every point (x, y), include (-x, y).

• Origin symmetry: For every point (x, y), include (-x, -y).

Example: The graph of y = |x| is symmetric about the y-axis.

Finding Symmetric Points

How to Find Points Symmetric to a Given Point

Given a point (a, b), its symmetric points with respect to various axes are:

• Example: The point (3, -5) has the following symmetric points:

  • With respect to the origin: (-3, 5)

  • With respect to the x-axis: (3, 5)

  • With respect to the y-axis: (-3, -5)

Additional info: These transformations are useful in graphing and analyzing the behavior of functions under reflection.