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Concept

Even Function

An even function is a function for which the value of f(-x) is equal to the value of f(x) for all the values in its domain. That is, opposite inputs have the same output.

f(- x) = f(x)

The graph of an even function is symmetric about the y-axis. The functions y=x^2 and y=2|x| are two examples of even functions.

Graphs of f(x)=x^2 and graph of g(x)=2|x|

Notice that if a function is even and the point (x,y) is on the graph, then the point (-x,y) is also on the graph. A function can be even, odd, or neither.

Extra

Determining If a Function is Even

To determine algebraically whether a function is even, substitute -x into the function rule and simplify. If the resulting expression is equal to f(x), then the function is even; otherwise, it is not. For example, consider the following function. f(x) = 3x^4 - 2x^2 + 1 Substitute -x for x and simplify.

f(x) = 3x^4 - 2x^2 + 1

Substitute

x= -x

f( -x) = 3( -x)^4 - 2( -x)^2 + 1

NegBaseToPosPow

(- a)^4 = a^4

f(-x) = 3x^4 - 2(-x)^2 + 1

NegBaseToPosBase

(- a)^2=a^2

f(-x) = 3x^4 - 2x^2 + 1

Substitute

3x^4 - 2x^2 + 1= f(x)