Problem 103 - College Algebra

Question

If you are given a function's graph, how do you determine if the function is even, odd, or neither?

Step-by-Step Solution

Answer

To determine if a graph represents an even, odd, or neither function, check for symmetry. If the graph is symmetrical about the y-axis, it's even. If it's symmetrical about the origin, it's odd. If it's not symmetrical about either the y-axis or the origin, it's neither.

Step 1: Identify symmetrical properties

First, inspect the graph visually to identify its symmetrical properties.

Step 2: Check for symmetry about the y-axis

Examine whether the function is symmetrical about the y-axis. This means the left and right halves of the graph are mirror images of each other. If this is the case, it's an even function.

Step 3: Check for symmetry about the origin

Examine whether the function is symmetrical about the origin. In this case, the graph in the second quadrant is a reflection of the graph in the fourth quadrant across the y-axis, and the graph in the third quadrant is a reflection of the graph in the first quadrant across the y-axis. If this is true, the function is odd.

Step 4: Check for no symmetry

If the function's graph is neither symmetrical about the y-axis nor about the origin, then the function is neither even nor odd.