Even Function

Definition:

An even function is a function that satisfies the condition: $$ f(x) = f(-x) $$ for all 𝑥 in its domain. This means that the function is symmetric about the y-axis. If you reflect the graph of an even function across the y-axis, it remains unchanged.

How to Recognize an Even Function:

1. Algebraically: Substitute − 𝑥 into the function and check if it results in the same function.
2. Graphically: The function should be symmetric with respect to the y-axis.

Examples:

1. $$ f(x) = x^2 $$ $$ f(-x) = (- x)^2 = x^2 = f(x) $$


2. $$ f(x) = x^4 + 2x^2 + 5 $$ $$ f(-x) = (- x)^4 + 2 (-x)^2 + 5 = x^4 + 2x^2 + 5 = f(x) $$


3. $$ f(x) = cos(x) $$ $$ f(-x) = cos(-x) = cos(x) = f(x) $$

Dynamic Examples:

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