Solved Examples on Even Function

Example 1: Check whether the function f(x) = x²+ 2x is even or not?

Solution:

For an even function, f(-x) = f(x)

We have, f(x) = x² + 2x

Now, f(-x) = (-x)² + 2(-x) = x² – 2x

We see that, f(-x) ≠ f(x)

Thus, f(x) is not an even function.

Example 2: State whether f(x) = x² + cos(x) is an even function or not?

Solution:

We have, f(x) = x² + cos(x)

We know that, x² and cos(x) are even functions. Also, addition of two even functions is even.

So, the given function f(x) = x² + cos(x) is an even function.

Example 3: Consider the function, f(x) = e^2x. State whether f(x) is even or not?

Solution:

We have, f(x) = e^2x

On putting -x in place of x, we get,

⇒ f(-x) = e^2(-x) = e^-2x = 1/e^2x

⇒ We have, e^2x ≠ 1/e^2x

Thus, f(-x) ≠ f(x)

Hence, f(x) is not an even function.

Example 4: Determine whether the function, f(x) = x⁴ tan²(x) is even or not?

Solution:

Here, we have product of two functions in f(x), i.e. x⁴ and tan²(x).

Let, g(x) = x⁴ and h(x) = tan²(x)

Substituting -x in g(x), we get,

g(-x) = (-x)⁴ = x⁴ = g(x)

Thus, g(x) is an even function.

Similarly for h(x), we have,

h(-x) = tan²(-x) = (-tan(x))² = tan²x = h(x)

Hence, h(x) is also an even function.

As, product of two even functions is an even function, we get f(x) is also an even function.

Even function is defined as a function that follows the relation f(-x) equals f(x), where x is any real number. Even functions have the same range for positive and negative domain variables. Due to this, the graph of even functions is always symmetric about the Y-axis in cartesian coordinates.

In this article, we will learn about even functions, their examples, properties, graphical representation of even functions, some solved examples, and practice questions related to even functions.