Which Of The Following Is An Even Function

Which of the Following is an Even Function? A Comprehensive Guide

Determining whether a function is even, odd, or neither is a fundamental concept in mathematics, particularly in calculus and advanced algebra. Understanding this concept is crucial for simplifying calculations, solving equations, and analyzing the symmetry of graphs. This article will provide a thorough explanation of even functions, how to identify them, and delve into various examples and exercises to solidify your understanding.

What is an Even Function?

A function is considered even if it satisfies a specific symmetry condition: f(-x) = f(x) for all x in the domain of the function. In simpler terms, if you replace x with -x and the function remains unchanged, it's an even function. This symmetry manifests graphically as a reflection symmetry about the y-axis. The graph of an even function looks identical when reflected across the y-axis.

Key characteristics of even functions:

• Symmetry about the y-axis: This is the most visually recognizable characteristic.
• f(-x) = f(x): This is the algebraic definition, and it's the definitive test.
• Often involves even powers of x: Terms like x², x⁴, x⁶, etc., contribute to even functions, as these terms remain unchanged when x is replaced by -x.

Identifying Even Functions: A Step-by-Step Approach

Let's break down the process of identifying even functions using a series of steps:

1. Substitute -x for x: This is the crucial first step. Replace every instance of x in the function with -x.

2. Simplify the Expression: Simplify the resulting expression using the rules of algebra. Remember that (-x)² = x², (-x)⁴ = x⁴, and so on. Even powers of negative x remain positive.

3. Compare with the Original Function: Compare the simplified expression from step 2 with the original function, f(x).

4. Determine Even or Not: If the simplified expression is identical to the original function (f(-x) = f(x)), then the function is even. If they are different, the function is not even. It could be odd or neither.

Examples of Even Functions

Let's illustrate the process with some examples:

Example 1: f(x) = x²

1. Substitute -x for x: f(-x) = (-x)²

2. Simplify: f(-x) = x²

3. Compare: f(-x) = f(x)

4. Conclusion: f(x) = x² is an even function. Its graph is a parabola symmetrical about the y-axis.

Example 2: f(x) = cos(x)

The cosine function is a classic example of an even function. Let's verify this:

1. Substitute -x for x: f(-x) = cos(-x)

2. Simplify: f(-x) = cos(x) (Remember the property: cos(-x) = cos(x))

3. Compare: f(-x) = f(x)

4. Conclusion: f(x) = cos(x) is an even function.

Example 3: f(x) = x⁴ - 3x² + 5

1. Substitute -x for x: f(-x) = (-x)⁴ - 3(-x)² + 5

2. Simplify: f(-x) = x⁴ - 3x² + 5

3. Compare: f(-x) = f(x)

4. Conclusion: f(x) = x⁴ - 3x² + 5 is an even function. Notice how all the terms involve even powers of x.

Example 4: f(x) = x³ + 2x

1. Substitute -x for x: f(-x) = (-x)³ + 2(-x)

2. Simplify: f(-x) = -x³ - 2x

3. Compare: f(-x) ≠ f(x)

4. Conclusion: f(x) = x³ + 2x is not an even function. In fact, this is an odd function (explained in the next section).

Distinguishing Even Functions from Odd Functions

It's important to distinguish even functions from odd functions. An odd function satisfies the condition f(-x) = -f(x). The graph of an odd function exhibits rotational symmetry about the origin. If a function doesn't satisfy either the even or odd function condition, it's classified as neither even nor odd.

Practical Applications of Even Functions

The concept of even functions has several practical applications across various fields:

• Physics: Many physical phenomena exhibit even symmetry. For example, the potential energy of a simple harmonic oscillator is an even function of displacement.
• Engineering: Even functions are used in signal processing and Fourier analysis to represent symmetric signals.
• Computer Graphics: Symmetry is a vital aspect of computer graphics, and even functions play a role in creating symmetrical shapes and patterns.

Advanced Concepts and Further Exploration

While the basic concept is straightforward, the application extends to more complex scenarios. For instance:

• Piecewise Functions: Determining whether a piecewise function is even requires checking the evenness condition for each piece of the function within its defined interval.
• Functions with Multiple Variables: The concept of even functions can be extended to functions with multiple variables. A function is even with respect to a variable if replacing that variable with its negative leaves the function unchanged.
• Taylor and Maclaurin Series: Even functions have Taylor series expansions that only contain even powers of x, while odd functions have Taylor series expansions with only odd powers of x.

Exercises to Test Your Understanding

Here are some exercises to practice identifying even functions:

1. f(x) = 4x⁶ - 7x⁴ + 2x² - 1
2. f(x) = |x|
3. f(x) = x³ - 5x
4. f(x) = 1/x²
5. f(x) = sec(x) (Recall trigonometric identities)
6. f(x) = e^x
7. f(x) = x² + 3x
8. f(x) = sin(x)
9. f(x) = 5 (A constant function)
10. f(x) = √(x²)

Solutions:

1. Even
2. Even
3. Odd
4. Even
5. Even
6. Neither
7. Neither
8. Odd
9. Even
10. Even

Conclusion

Identifying even functions is a crucial skill in mathematics and its applications. By understanding the definition, the step-by-step process, and the various examples, you can confidently determine whether a function is even, laying a solid foundation for more advanced mathematical concepts. Remember to always apply the key criterion: f(-x) = f(x). Practice consistently using the examples and exercises provided, and you'll master this essential concept in no time.