Problem 9 - Precalculus with Calculus Previews

Question

In Problems 9-12, determine whether the given polynomial function \(f\) is even, odd, or neither even nor odd. Do not graph. \(f(x)=-2 x^{3}+4 x\)

Step-by-Step Solution

Answer

The function \(f(x) = -2x^3 + 4x\) is odd.

Step 1: Define Even Function

A function is called even if for every real number \(x\), \(f(x) = f(-x)\). Evaluate the function to check this condition.

Step 2: Define Odd Function

A function is called odd if for every real number \(x\), \(f(-x) = -f(x)\). We will check this after evaluating \(f(-x)\).

Step 3: Evaluate \(f(x)\) at \(-x\)

Substitute \(-x\) into the polynomial. Calculate \(f(-x) = -2(-x)^3 + 4(-x) = -2(-x)^3 - 4x = 2x^3 - 4x\).

Step 4: Check Even Condition

Compare \(f(-x)\) to \(f(x)\). Since \(f(-x) = 2x^3 - 4x\) and \(f(x) = -2x^3 + 4x\), \(f(-x) eq f(x)\). Thus, the function is not even.

Step 5: Check Odd Condition

Check if \(f(-x) = -f(x)\). Here, \(-f(x) = -(-2x^3 + 4x) = 2x^3 - 4x\). Since \(f(-x) = 2x^3 - 4x\) matches \(-f(x)\), the function is odd.

Key Concepts

Even and Odd Functions

Understanding even and odd functions is crucial when working with polynomials. These concepts help determine the symmetry of graphs, even without plotting them.
Before diving in, let's clarify their definitions.

• Even functions: A function is even if the equation \( f(x) = f(-x) \) holds true for all \( x \). This indicates that the function's graph is symmetric with respect to the y-axis.
• Odd functions: A function is odd if \( f(-x) = -f(x) \). In this case, the graph will be symmetric about the origin.

Now, let's apply these definitions to a polynomial function, \( f(x) = -2x^3 + 4x \). When evaluating its symmetry, we first compute \( f(-x) \). By substituting \(-x\) into the function, we get \( f(-x) = 2x^3 - 4x \).
Comparing \( f(-x) \) to the original function \( f(x) \), we see they are not identical, confirming that it's not even.
Then, see if \( f(-x) = -f(x) \); if both sides match, as in this case, the function is odd.

Evaluating Functions

Evaluating functions involves substituting specific values into the function's expression to find the corresponding output. For polynomial functions, this means replacing the variable \( x \) with another number or expression.
In the case of determining if a function is even or odd, we substitute \(-x\) into the function. This process helps us ascertain the function's symmetry properties.

• Start by replacing \( x \) with \(-x\). For \( f(x) = -2x^3 + 4x \), substituting gives \( f(-x) = -2(-x)^3 + 4(-x) \).
• Simplify the expression: Calculate the output. Here, \(-2(-x)^3\) becomes \(2x^3\) and \(4(-x)\) simplifies to \(-4x\).
• Compare the results: Finally, compare \( f(x) \) and \( f(-x) \) to see if the conditions of being even or odd hold true.

This straightforward substitution and calculation reveal the symmetry type of the function.

Mathematical Definitions

Mathematical definitions are fundamental in guiding the correct evaluation of functions and understanding their properties. In the context of polynomial functions, knowing the definitions of even and odd functions is key.

• Even function: Defined by \( f(x) = f(-x) \). An even function's graph mirrors itself across the y-axis, indicating that substituting \(-x\) yields the same result as \(x\).
• Odd function: Characterized by \( f(-x) = -f(x) \). An odd function's graph rotates 180 degrees around the origin, showing a balanced symmetry that resembles flipping both vertically and horizontally.

These definitions guide us when conducting algebraic verification, such as evaluating \( f(x) \) and using substitution methods to decide the nature of a polynomial's symmetry.
Understanding these mathematical principles enables accuracy and confidence in solving problems involving polynomial functions.