Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The graph of a function with origin symmetry can rise to the left and rise to the right.

Name: Problem 107, Chapter 2: Precalculus
Uploaded: 2021-10-07T19:12:56-08:00
Duration: 1 min 21 s
Channel: Julie Buscema
Description: Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The graph of a function with origin symmetry can rise to the left and rise to the right.

   Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
The graph of a function with origin symmetry can rise to the left and rise to the right.

Precalculus

Robert Blitzer 6th Edition

Chapter 2, Problem 107

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Solved by Expert Julie Buscema

10/07/2021

Step 1

A function has origin symmetry if it is unchanged when rotated by 180° about the origin. In other words, for every point (x, y) on the graph, the point (-x, -y) is also on the graph. Show more…

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Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The graph of a function with origin symmetry can rise to the left and rise to the right.

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Key Concepts

Origin Symmetry

This concept refers to a graph being symmetric with respect to the origin, meaning that if (x, y) is on the graph, then (-x, -y) must also be on the graph. This symmetry implies a rotation of 180 degrees about the origin, and is a key characteristic of functions known as odd functions.

Odd Functions

An odd function is one that satisfies the condition f(-x) = -f(x) for every x in its domain. This property directly arises from the function’s origin symmetry, and it ensures that the behavior of the function on one side of the origin is the mirror image (with sign inversion) of its behavior on the other side.

Monotonicity

Monotonicity describes whether a function is consistently increasing or decreasing over an interval. For functions with origin symmetry (odd functions), if the function is increasing on one side of the origin, it must be decreasing on the other, because the transformation f(-x) = -f(x) reverses the direction of change.

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