Problem 61 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

The function is odd.

Step 1: Understand Definitions

A function is \textbf{even} if \( f(-x) = f(x) \) for all \( x \) in the domain. A function is \textbf{odd} if \( f(-x) = -f(x) \) for all \( x \) in the domain.

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

Substitute \(-x\) into the function \( f(x) = 3x^3 - x \).\[ f(-x) = 3(-x)^3 - (-x) \]

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

Simplify \( f(-x) = 3(-x)^3 - (-x) \).\[ f(-x) = 3(-x^3) + x = -3x^3 + x \]

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

Calculate \(-f(x)\) by multiplying the original function by -1.\[-f(x) = -(3x^3 - x) = -3x^3 + x\]Compare \( f(-x) = -3x^3 + x \) with \(-f(x) = -3x^3 + x \). Since \( f(-x) = -f(x) \), the function is odd.

Key Concepts

Function Substitution

Function substitution is a key step in determining whether a given function is even or odd. This involves taking the original function and substituting \(-x\) for every occurrence of \(x\). By doing this, we can analyze how the function behaves when we input the opposite of any value \(x\).

To apply substitution, consider the function:

• Given: \(f(x) = 3x^3 - x\)
• Substitute \(-x\) in place of \(x\): \(f(-x) = 3(-x)^3 - (-x)\)

This step is crucial as it sets the stage for simplification. It allows us to see the form of the function when the input is inverted, which is key to identifying the function's symmetry properties.
Smooth substitution ensures that we can proceed to the simplification stage with confidence.

Function Simplification

After substituting \(-x\) for \(x\), it is necessary to simplify the expression. Simplification helps in understanding how the function is transformed by the substitution process. It involves applying algebraic rules to make the expression more straightforward.

With the function \(f(-x) = 3(-x)^3 - (-x)\), we simplify step-by-step:

• Calculate \((-x)^3\), which yields \(-x^3\).
• This turns the expression into: \(f(-x) = 3(-x^3) + x\).
• Result: \(f(-x) = -3x^3 + x\).

By simplifying, we achieve a clear expression that we can then compare with the original function. This step is crucial because it often uncovers symmetry properties that might not be immediately obvious in the unsimplified expression.

Comparing Functions

The final step involves comparing the simplified version of \(f(-x)\) to the original function \(f(x)\) and its negation, \(-f(x)\). This comparison reveals the even or odd nature of the function.

• From our example: \(f(x) = 3x^3 - x\).
• Simplified function after substitution: \(f(-x) = -3x^3 + x\).
• Negate the original function: \(-f(x) = -(3x^3 - x) = -3x^3 + x\).

Upon comparing \(f(-x)\) and \(-f(x)\), we see they are identical. This means \(f(-x) = -f(x)\), indicating an odd function.

Comparing these functions allows us to confidently determine whether the function behaves symmetrically about the y-axis (even) or origin (odd). This method provides a robust framework for identifying function properties.