Even Functions : Definition , Graph ,examples

What is even functions

An even function is a function f(x) that satisfies the property f(x) = f(-x) for all x in the domain of the function.
In other words, an even function is symmetric with respect to the y-axis.
Geometrically, an even function has the property that if you fold the graph of the function along the y-axis, the two halves of the graph overlap perfectly.
Graph of even function is symmetrical in I and II or III and IV quadrants

Examples

(1) Polynomial functions

Any polynomial with only even degree terms is an even function. For example, f(x) = 6x⁸ – 6x² – 5.

f(-x)= 6(-x)^8 -6(-x)^2 – 5 =6x^8 -6x^2 -5= f(x)

(2) Cosine Function [cos(x)]

The cosine function is defined as the ratio of the adjacent side to the hypotenuse in a right-angled triangle. It is an even function because cos(x) = cos(-x) for all x. This property can also be derived from the geometric interpretation of cosine, which is that it represents the x-coordinate of a point on the unit circle.

(3) Parabola Function: $x^2$

The parabola function is defined as f(x) = x^2. It is an even function because f(x) = f(-x) for all x. This property can be seen by observing that the graph of the function is symmetric about the y-axis

(4) Absolute Value Function: |x|

The absolute value function is defined as f(x) = |x|. It is an even function because f(x) = f(-x) for all x. This can be seen by using the definition of absolute value, which is that |x| is equal to x when x is positive and -x when x is negative.

Properties of even function

For any function f(x), f(x) + f(-x) is an even function.
The sum or difference of two even functions is even.

let f(x)= f(-x) and g(x) = g(-x)

Now h(x) = f(x) + g(x)
h(-x) = f(-x) +g(-x) =f(x) + g(x)= h(x)

The multiple of an even function is again an even function.

let f(x)= f(-x) and h(x) = n f(x)
then
h(-x) = nf(-x) = nf(x) = h(x)

The product or division of two even functions is even.

let f(x)= f(-x) and g(x) = g(-x)

Now h(x) = f(x) g(x)
h(-x) = f(-x) g(-x) =f(x) g(x)= h(x)

The composition of two even functions and the composition of an even and odd function is even.

let Both even f(x)= f(-x) and g(x) = g(-x)

$f \circ g (x) = f(g(x)$
Now $f \circ g (-x)= f(g(-x)= f(g(x)$

let one even f(x)= f(-x) and g(x) =- g(-x)

$f \circ g (x) = f(g(x))$
Now $f \circ g (-x)= f(g(-x)= f(-g(x))=f(g(x))$

How to Prove the even Function

We simply need to prove that f(x) = f(-x)

Solved Examples

Example 1

$f(x) = \cos (sin^3 x)$

Solution

$f(-x)= \cos (sin^3 -x)= \cos [(-sin x)^3]= \cos (- \sin^3 x) = \cos (sin^3 x) = f(x)$

So it is even function

Example 2

f(x) = x² + 3x – 6