91Ó°ÊÓ

An even function has a special symmetry. This means that the function behaves the same when you replace x with -x. Mathematically, a function \(f(x)\) is even if \(f(-x) = f(x)\) for every x in the domain.
A classic example is the function \( f(x) = x^2 \). If you calculate \( f(-x) \), you get \( (-x)^2 \), which is just \( x^2 \) again. Hence, \( f(x) = x^2 \) is even because \( f(-x) = f(x) \).
Even functions often have graphs that are symmetrical about the y-axis. Think about the graph of \( y = x^2 \). If you fold it along the y-axis, both halves match exactly.
So, whenever you are given a function to check if it is even, re-calculate by substituting -x and see if you get back the same function.

Odd functions have a unique type of symmetry. When you replace x with -x, the function should change to its negative self. In other words, \(f(x)\) is odd if \(f(-x) = -f(x)\) for all x in the domain.
A simple example is \( f(x) = x^3 \). If you calculate \( f(-x) \), you get \( (-x)^3 = -x^3 \), which is the negative of \( f(x) \). Thus, \( f(x) = x^3 \) is an odd function because \( f(-x) = -f(x) \).
The graph of an odd function is symmetric about the origin. Imagine the graph of \( y = x^3 \). If you rotate it 180 degrees around the origin, it looks the same.
When checking if a function is odd, substitute -x and see if the result is the negative of the original function.

Function analysis involves understanding the behavior and properties of a function. For even and odd functions, you're specifically looking for symmetry.
Start by substituting -x into the function to determine \( f(-x) \).
Then compare \( f(-x) \) to \( f(x) \) and -\( f(x) \).

• If \( f(-x) = f(x) \), the function is even.
• If \( f(-x) = -f(x) \), the function is odd.
• If neither condition is true, the function is neither even nor odd.

This process is key to understanding the function's symmetry and categorizing it correctly.

Algebraic verification is the process of using algebraic manipulations to determine if a function is even, odd, or neither. It's a step-by-step method to ensure mathematical accuracy.
Let's use the example given: \( h(x) = \frac{x}{x^2 - 1} \)
1. Find \( h(-x) \): Substitute -x for x.
\( h(-x) = \frac{-x}{(-x)^2 - 1} = \frac{-x}{x^2 - 1} \)
2. Compare \( h(x) \) and \( h(-x) \):
We see that \( h(-x) = \frac{-x}{x^2 - 1} \), which is -\( h(x) \).
3. Since \( h(-x) = -h(x) \), we conclude that \( h(x) \) is an odd function.
This careful substitution and comparison confirm the function's category and assure us of the solution's accuracy.