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Chapter 2: Problem 107

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.

Short Answer

Expert verified
The statement is false. The correct statement would be: 'The graph of a function with origin symmetry can rise to the left and fall to the right, or fall to the left and rise to the right.'

Step by step solution

01

Identify the Problem

The statement in the exercise is 'The graph of a function with origin symmetry can rise to the left and rise to the right.' It needs to be verified as true or false.
02

Analyze the Definition of Origin Symmetry

A function has origin symmetry if its graph remains the same upon being rotated 180 degrees about the origin. Mathematically, this occurs when f(-x) = -f(x) for every x in the function's domain. This means that if the graph on one side of the origin (say the left side) rises, then the graph on the other side (the right side) must fall, and vice versa. It cannot rise on both sides.
03

Assess the Given Statement

The given statement suggests that a function with origin symmetry can rise to the left and right of the origin. This is contrary to the nature of origin symmetry as analyzed in Step 2.
04

Correct the Statement

The correct statement should be 'The graph of a function with origin symmetry can rise to the left and fall to the right, or fall to the left and rise to the right.' This statement describes accurately the behavior of graphs of functions with origin symmetry.

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