Problem 30 - Calculus

Question

Are the functions even, odd, or neither? $$f(x)=x^{4}-x^{2}+3$$

Step-by-Step Solution

Answer

The function is even.

Step 1: Understand Even and Odd Functions

A function is *even* if it satisfies the condition \( f(x) = f(-x) \) for all \( x \) in the domain. A function is *odd* if it satisfies the condition \( f(-x) = -f(x) \) for all \( x \) in the domain. If neither condition is met, the function is neither even nor odd.

Step 2: Substitute \( -x \) into \( f(x) \)

Let's find \( f(-x) \). Replace \( x \) with \( -x \) in the function definition: \(f(x) = x^4 - x^2 + 3 \)\(f(-x) = (-x)^4 - (-x)^2 + 3 \).

Step 3: Simplify \( f(-x) \)

Calculate the simplified result of \( f(-x) \):\((-x)^4 = x^4\),\((-x)^2 = x^2\).Therefore, \( f(-x) = x^4 - x^2 + 3 \).

Step 4: Compare \( f(x) \) and \( f(-x) \)

Now, compare \( f(x) \) with \( f(-x) \):\( f(x) = x^4 - x^2 + 3 \)\( f(-x) = x^4 - x^2 + 3 \).Since \( f(x) = f(-x) \), the function \( f(x) \) is even.

Key Concepts

Even Functions

Even functions have a particular symmetry about the y-axis, which makes them quite intriguing. The primary characteristic of an even function is that substituting a negative input

• yields the same output as the positive input.
• This can be expressed mathematically as: \( f(x) = f(-x) \).

One way to visualize this is to think of folding the graph along the y-axis — if both halves match perfectly, the function is even.
Consider the function \( f(x) = x^4 - x^2 + 3 \). When we substitute \(-x\) in place of \(x\), it results in \( f(-x) = x^4 - x^2 + 3 \), which is identical to \( f(x) \). This verification confirms that the function is indeed even. The power of symmetry makes computations and analyses with even functions very straightforward and often predictable.

Odd Functions

Odd functions have a distinct kind of symmetry — rotational symmetry around the origin. This means that if you rotate the graph by 180 degrees, it will look exactly the same. The defining characteristic of odd functions is expressed mathematically as:

• \( f(-x) = -f(x) \).

This means that for an odd function:

• The output for a negative input is the negative of the output for a positive input.

This property often results in graphs that appear to have a twisted symmetry about the origin.
It's important to note that not every function is either even or odd; some don't satisfy either condition. For a function like \( f(x) = x^4 - x^2 + 3 \), since it does not satisfy \( f(-x) = -f(x) \), it cannot be classified as odd.

Function Symmetry

Function symmetry is all about how the graph of a function mirrors itself. Knowing whether a function is even or odd can greatly simplify both solving equations and understanding the function's behavior. Symmetry helps in recognizing repeating patterns in functions, which is key in calculus and many other fields of mathematics.

• Even functions reflect symmetrically across the y-axis.
• Odd functions exhibit rotational symmetry around the origin.

When you start investigating a function, testing for symmetry can give you powerful insights. This is especially useful in integration and other complex calculations, where symmetrical properties can reduce the workload significantly.
Having a grasp on symmetry plays into more advanced calculus concepts, such as Fourier series, where symmetry directly impacts the components you'll end up working with. So, whether you're solving equations or analyzing graphs, identifying symmetry can be a very powerful tool at your disposal.