Problem 106 - 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 function is even, check if its graph is symmetric about the y-axis. For checking if it is odd, observe if it is symmetric about the origin. If the graph isn't symmetric around either, the function is neither even nor odd.

Step 1: Identify Even Functions

Check if the function graph is symmetric around the y-axis. If it is, the function is said to be an even function. This is based on the mathematical definition: a function is even if and only if for any x and -x in the domain, \(f(x) = f(-x)\). A visual representation of this is symmetry about the y-axis.

Step 2: Identify Odd Functions

Check if the function's graph is symmetric around the origin. If it is, then the function is said to be an odd function. Mathematically speaking, a function is odd if and only if for any x and -x in the domain, \(f(x) = -f(-x)\). A visual representation of this is symmetry about the origin.

Step 3: Identify Neither

If the function's graph is not symmetric about the y-axis or the origin, then the function is neither even nor odd.