Problem 37 - Precalculus : Building Concepts and Connections

Question

Decide if each function is odd, even, or neither by using the definitions. $$f(x)=-x^{3}+1$$

Step-by-Step Solution

Answer

The function \(f(x) = -x^3 + 1\) is an odd function.

Step 1: Compute f(-x)

The first step is to substitute \(-x\) for \(x\) in the function. This gives \(f(-x) = -(-x)^3 + 1 = -x^3 + 1\).

Step 2: Compare f(-x) with f(x) and -f(x)

We see that \(f(-x) = -x^3 + 1\) is not equal to \(f(x) = -x^3 + 1\), so the function cannot be even. However, we see that \(f(-x) = -f(x)\), which satisfies the condition for odd functions.

Step 3: Determine the type of function

Since \(f(-x) = -f(x)\), the function \(f(x) = -x^3 + 1\) is an odd function.

Key Concepts

Function Symmetry

When we talk about function symmetry, we primarily discuss even and odd functions. Function symmetry is essential in mathematics because it helps to understand the behavior and properties of a function.

**Even Functions**: These functions are symmetric with respect to the y-axis. For a function to be even, the following must hold true:

• The condition \( f(-x) = f(x) \) applies for all x in the domain.
• A classic example of an even function is \( f(x) = x^2 \).

**Odd Functions**: Odd functions, on the other hand, are symmetric with respect to the origin. This leads to a particular property:

• For a function to be odd, \( f(-x) = -f(x) \) should be true for all x in the domain.
• An example of an odd function is \( f(x) = x^3 \).

Understanding these symmetries can simplify graphing functions and solving equations. Try plotting both function types to visually see the symmetry!

Polynomial Functions

Polynomial functions are mathematical expressions involving a sum of powers in one or more variables. Each term in a polynomial function consists of a coefficient, a variable, and an exponent.

• A polynomial is typically written in the form \( a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \), where \( a_n, a_{n-1}, \ldots, a_0 \) are constants.
• In the simplest case, such as\( f(x) = -x^3 + 1 \), we see a polynomial of degree 3.

Polynomials can exhibit characteristics of odd or even functions depending on their constituent terms:

• Polynomial functions with all even degree terms are usually even functions.
• If all the terms have odd degrees, they're typically odd functions.

In our exercise, the function \( f(x) = -x^3 + 1 \) includes both odd and constant terms; however, its behavior skews towards being an odd function due to the highest-degree term being odd.

Mathematical Proofs

Understanding mathematical proofs is crucial when verifying the nature of a function's symmetry. Proofs give us a step-by-step logic that confirms whether a function is even, odd, or neither.

Let's examine the proof for odd functions, using our original function \( f(x) = -x^3 + 1 \) as an example:

• We start by computing \( f(-x) \), replacing \( x \) with \( -x \), resulting in \( -(-x)^3 + 1 = -x^3 + 1 \).
• Compare this with \( f(x) \). Here, you'll notice \( f(-x) = f(x) \) doesn't hold, indicating it's not even.
• Next, observe that \( f(-x) = -(-x^3 + 1) = -f(x) \), satisfying the condition for odd functions.

By carefully following these proof steps, you can confidently determine a function's symmetry. This logical process reinforces mathematical understanding and validates your solution with certainty.