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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) = 3 x^{4} - 2 x^{2} + 1

Substitute

- x

for

x

and simplify.

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

Substitute

$x = - x$

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

NegBaseToPosPow

$(- a)^{4} = a^{4}$

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

NegBaseToPosBase

$(- a)^{2} = a^{2}$

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

Substitute

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

f (- x) = f (x)

Since

f (- x) = f (x),

the given function is even.

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