How to determine if a function is even or odd

English
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how to determine if a function is even or odd

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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.

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

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

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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!