Problem 102 - College Algebra

Question

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

Step-by-Step Solution

Answer

To determine if a function is even, substitute \( -x \) for \( x \) in the function's equation, and if the result is the same as the original function, it's even. If the result is the negative of the function, it's odd. If the result is neither, the function is neither even nor odd.

Step 1: Definition of Even and Odd Functions

Even function definition: A function \( f(x) \) is even if \( f(-x) = f(x) \), for all values of \( x \) in the function's domain. Meaning the function has symmetry around the y-axis. \n Odd function definition: A function \( f(x) \) is odd if \( f(-x) = -f(x) \), for all values of \( x \) in the function's domain. This means the function has symmetry about the origin.

Step 2: Apply the Definition to the Function

Substitute \( x \) with \( -x \) in the function's equation. This is the value of \( f(-x) \).

Step 3: Compare the Resultant Equation with the Original Function

If the resultant equation from Step 2 is the same as the original function, the function is even. If the resultant equation is the negative of the function, then the function is odd.

Step 4: Determine If Function is Even, Odd, or Neither

Based on the comparisons made in Step 3, you can now categorize the function. If there are no matches from Step 3, then the function is neither even nor odd.