Problem 103 - College Algebra Essentials

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

First, check if the graph is symmetric around the y-axis. If so, it's an even function. If not, see if it is rotationally symmetric around the origin. If so, it's an odd function. If it fails both checks, it's neither an odd nor an even function.

Step 1: Check for Symmetry Around the Y-Axis

To begin, check if the graph of the function is symmetrical around the y-axis. The graph's left side must mirror its right side. Draw a vertical line at \( x = 0 \). If it acts as a line of symmetry (i.e., splits the graph into two identical halves), then the function is even.

Step 2: Check for Rotational Symmetry Around the Origin

If the first check does not classify the function, next check for rotational symmetry about the origin. To do this, rotate the graph 180 degrees about the origin. If it remains the same, it means that the function is odd. This implies that every point \((x, y)\) on the graph gets rotated to \((-x, -y)\).

Step 3: Classify as Neither

If the graph exhibits neither y-axis symmetry nor origin-rotational symmetry, we can classify the function as neither odd nor even.