Problem 66 - A Graphical Approach to Precalculus with Limits

Question

Each function is either even or odd Evaluate \(f(-x)\) to determine which situation applies. $$f(x)=x^{3}-4 x$$

Step-by-Step Solution

Answer

The function is odd.

Step 1: Understand the Problem

We have a function \(f(x) = x^3 - 4x\). The task is to determine if this function is even, odd, or neither by evaluating \(f(-x)\) and comparing it to \(f(x)\) and \(-f(x)\).

Step 2: Calculate \(f(-x)\)

Substitute \(-x\) into the function: \(f(-x) = (-x)^3 - 4(-x)\). Simplify this expression to find \(-x^3 + 4x\).

Step 3: Compare \(f(-x)\) to \(f(x)\)

Compare \(f(-x) = -x^3 + 4x\) to \(f(x) = x^3 - 4x\). Clearly, \(f(-x) ot= f(x)\), which means the function is not even.

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

Calculate \(-f(x) = -(x^3 - 4x) = -x^3 + 4x\). Note that \(f(-x) = -f(x)\), which means the function is odd.

Key Concepts

Even and Odd Functions

In the world of precalculus, understanding whether a function is even or odd is crucial in analyzing its symmetry. Let's dive into what makes a function even, odd, or neither.

An **even function** is symmetric about the y-axis. Mathematically, a function \(f(x)\) is even if \(f(-x) = f(x)\). This means when you substitute \(-x\) into the function, the output remains the same as when you input \(x\).

On the other hand, an **odd function** is symmetric about the origin. A function is considered odd if \(f(-x) = -f(x)\). So, replacing \(x\) with \(-x\) changes the output to the negative of the original function. Simply put:

• If \(f(-x) = f(x)\), the function is even.
• If \(f(-x) = -f(x)\), the function is odd.
• If neither condition is satisfied, the function is neither even nor odd.

In the exercise, substituting \(-x\) into the function gave us \(f(-x) = -x^3 + 4x\), which is precisely the negative of \(f(x) = x^3 - 4x\). Thus, it confirms that our function is odd.

Function Evaluation

Evaluating a function means finding the output for a given input. This is a vital skill in mathematics as it helps us understand the behavior of the function under different conditions.

When given a function, like \(f(x) = x^3 - 4x\), you might be asked to evaluate it at a specific point, say \(x = 2\). To do this, substitute 2 into the function in place of \(x\):
1. Calculate \(f(2) = 2^3 - 4(2)\)
2. Simplify the expression to find \(f(2) = 8 - 8 = 0\)

This process is straightforward but essential. It involves:

• Substitution: replacing the variable with the given number.
• Simplification: executing any operations needed to find a numerical result.

Function evaluation is not only necessary for basics but also helps in advanced topics, such as finding function roots and understanding transformations.

Polynomial Functions

Polynomial functions are a staple in precalculus. They are expressions consisting of variables raised to various powers, and they come with coefficients. A polynomial function is of the form \(a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\). The exponents (\(n\)) are whole numbers, and the coefficients (\(a_n\)) are real numbers.
Our example function \(f(x) = x^3 - 4x\) fits seamlessly into this definition. It is a polynomial of degree 3 because the highest power of x is 3. Its structure allows us to:

• Identify terms based on powers of \(x\).
• Determine end behavior of the graph by looking at the highest degree.
• See that there are infinite points plotted, following a predictable pattern.

Polynomial functions can be complex but are key to many calculus concepts like limits and derivatives. Each polynomial can be analyzed for roots, where the graph intersects the x-axis, among other features. Understanding these fundamentals makes exploring advanced concepts more approachable.