How to tell if a function is even odd or neither?

A function is defined as the change in the output value with respect to input where the output variable is dependent upon the input variable.

Answer: For an even function, f(-x) = f(x), for all x, for an odd function f(-x) = -f(x), for all x. If f(x) ≠ f(−x) and −f(x) ≠ f(−x) for some values of x, then f is neither even nor odd.

Let's understand the solution.

Explanation:

(a) Let's understand even functions.

If a given function is symmetric about the y-axis, it is known as an even function.

A function is even if f(x) = f(−x) for all values of x

For an even function f(x), if we plug in −x in place of x, then the value of f(−x) is equal to the value of f(x).

Thus, the formula to check if the function is even is given as:

f(x) = f(−x)

Let's take an example to understand even functions.

Example: f(x) = x²

f(-x) = (-x)² = x²

Thus, we see that f(x) = f(-x)

Hence, the given function f(x) = x² is an even function.

Let's check it graphically.

We see that the graph of y = x² is symmetric about the y-axis and hence an even function.

(b) Let's look into odd functions:

A function in which one side of the x-axis is sign-inverted with respect to the other side or graphically, symmetric about the origin is known as an odd function.

A function is odd if f(-x) = - f(x) for all values of x

For an odd function f(x), if we plug in −x in place of x, then the value of f(−x) is equal to the value of - f(x).

Thus, the formula to check if the function is odd is given as:

f(-x) = - f(x)

Let's take an example to understand odd functions.

Example: f(x) = x³

f(-x) = (-x)³ = - x³

Also, - f(x) = - x³

Thus, f(x) = x³ is an odd function since, f(-x) = - f(x).

Let's check it graphically.

The graph looks symmetrical about the origin, thus it is an odd function.

(c) Let's understand about the function that is neither even nor odd

A function f(x) in which f(x) ≠ f(−x) and −f(x) ≠ f(−x) for any value of x is neither an even function, nor an odd function.

Graphically, these functions are neither symmetric about origin nor about the y-axis.

Let's take an example to understand this.

Example: f(x) = 2x⁵ + 3x² + 1

f(-x) = - 2x⁵ + 3x² + 1

- f(x) = - 2x⁵ - 3x² - 1

Thus, we see that f(x) ≠ f(−x) and −f(x) ≠ f(−x) for the given function. Hence, it is neither even nor an odd function.

Let's look at the graph of this function.

We see that the graph is neither symmetric about origin nor about the y-axis. Thus, it is neither even nor an odd function.

Therefore, we have understood about even, odd, and neither even nor odd functions.