Understanding Origin Symmetry in Math: Definition and Examples

Function symmetric through the origin

A function is said to be symmetric through the origin if it possesses a special property called origin symmetry or odd symmetry

A function is said to be symmetric through the origin if it possesses a special property called origin symmetry or odd symmetry.

A function f(x) is symmetric through the origin if substituting -x for x in the function yields the negative of its original value, i.e., f(-x) = -f(x) for all x in the domain of the function.

Mathematically, if f(x) is a function symmetric through the origin, then f(-x) = -f(x) for all x in the domain of the function.

To demonstrate this using an example, let’s consider the function f(x) = 3x.

To check if f(x) is symmetric through the origin, we substitute -x for x in the function:

f(-x) = 3(-x) = -3x

Now, let’s compare this with the negative of the original function:

-f(x) = -3x

Since f(-x) = -f(x), we can conclude that the function f(x) = 3x is symmetric through the origin.

Symmetric functions through the origin have certain characteristics:

1. They pass through the origin: The point (0, 0) is always on the graph of a function symmetric through the origin.

2. They are odd functions: Functions that are symmetric through the origin are classified as odd functions. Odd functions have the property that f(-x) = -f(x).

3. Graphical representation: Symmetric functions through the origin have a symmetrical appearance with respect to the origin. If any point (x, y) is on the graph, then (-x, -y) will also be on the graph.

Remember that not all functions are symmetric through the origin. Functions that are even symmetric or have no symmetry through the origin are quite common as well.

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