# How to determine if a function is even or odd

how to determine if a function is even or odd

how to determine if a function is even or odd

Answer: To determine if a function is even or odd, you can use the properties of even and odd functions:

1. Even Function:

• A function f(x) is even if and only if it satisfies the following property: f(x) = f(-x) for all x in its domain.
• Geometrically, even functions have symmetry about the y-axis; if you reflect the graph across the y-axis, it remains unchanged.
• Common examples of even functions include f(x) = x^2 and f(x) = cos(x).
2. Odd Function:

• A function f(x) is odd if and only if it satisfies the following property: f(x) = -f(-x) for all x in its domain.
• Geometrically, odd functions have rotational symmetry of 180 degrees about the origin; if you rotate the graph by 180 degrees, it remains unchanged.
• Common examples of odd functions include f(x) = x^3 and f(x) = sin(x).

To determine if a function is even or odd, you can perform the following steps:

1. Replace x with -x in the function.
2. Simplify the expression.
3. Compare the original function with the expression obtained after replacing x with -x:
• If f(x) = f(-x), the function is even.
• If f(x) = -f(-x), the function is odd.
• If neither of these conditions holds, the function is neither even nor odd.

Keep in mind that not all functions are classified as even or odd; some are neither. Additionally, some functions may have even or odd parts, meaning they can be decomposed into both even and odd components.

Example 1: Even Function
Function: f(x) = x^4 - x^2

1. Replace x with -x in the function:
f(-x) = (-x)^4 - (-x)^2 = x^4 - x^2

2. Simplify the expression:
f(-x) = x^4 - x^2

3. Compare the original function with the expression after replacing x with -x:
f(x) = x^4 - x^2
f(-x) = x^4 - x^2

In this case, f(x) = f(-x), so the function f(x) = x^4 - x^2 is an even function because it satisfies the property of even functions. It has symmetry about the y-axis.

Example 2: Odd Function
Function: g(x) = x^3 - x

1. Replace x with -x in the function:
g(-x) = (-x)^3 - (-x) = -x^3 + x

2. Simplify the expression:
g(-x) = -x^3 + x

3. Compare the original function with the expression after replacing x with -x:
g(x) = x^3 - x
g(-x) = -x^3 + x

In this case, g(x) = -g(-x), so the function g(x) = x^3 - x is an odd function because it satisfies the property of odd functions. It has rotational symmetry of 180 degrees about the origin.

Example 2: Even Function
Function: h(x) = 2x^2 + 3x^4

1. Replace x with -x in the function:
h(-x) = 2(-x)^2 + 3(-x)^4 = 2x^2 + 3x^4

2. Simplify the expression:
h(-x) = 2x^2 + 3x^4

3. Compare the original function with the expression after replacing x with -x:
h(x) = 2x^2 + 3x^4
h(-x) = 2x^2 + 3x^4

In this case, h(x) = h(-x), so the function h(x) = 2x^2 + 3x^4 is an even function because it satisfies the property of even functions. It has symmetry about the y-axis.

Of course! Here’s another example in English with mathematical expressions represented using the “\$” symbol:

Example 3: Odd Function
Function: k(x) = 5x^3 - 2x

1. Replace x with -x in the function:
k(-x) = 5(-x)^3 - 2(-x) = -5x^3 + 2x

2. Simplify the expression:
k(-x) = -5x^3 + 2x

3. Compare the original function with the expression after replacing x with -x:
k(x) = 5x^3 - 2x
k(-x) = -5x^3 + 2x

In this case, k(x) = -k(-x), so the function k(x) = 5x^3 - 2x is an odd function because it satisfies the property of odd functions. It has rotational symmetry of 180 degrees about the origin.

how to determine if a function is even or odd? Please example @sorumatikbot

Certainly! Let’s use an example to determine if a function is even or odd.

Example:
Consider the function f(x) = x^2 - 2.

To determine if this function is even or odd, we need to compare f(x) with f(-x).

1. Evaluating f(x):
f(x) = x^2 - 2

2. Evaluating f(-x):
f(-x) = (-x)^2 - 2 = x^2 - 2

3. Comparing f(x) with f(-x):
f(x) = x^2 - 2
f(-x) = x^2 - 2

In this case, f(x) = f(-x), which means the function f(x) is even. It has symmetry about the y-axis.

So, based on this comparison, we can conclude that the function f(x) = x^2 - 2 is an even function.

If you have any further questions or need additional examples, feel free to ask!