Problem 73 - A Graphical Approach to Precalculus with Limits

Question

Each function is either even or odd Evaluate \(f(-x)\) to determine which situation applies. $$f(x)=\sqrt[3]{x^{3}-5 x}$$+

Step-by-Step Solution

Answer

The function is odd.

Step 1: Understand the Problem

To determine if a function is even, odd, or neither, we need to evaluate the function at \(-x\) and compare it to the original function \(f(x)\). An even function satisfies \(f(-x) = f(x)\), while an odd function satisfies \(f(-x) = -f(x)\).

Step 2: Evaluate f(-x)

We need to substitute \(-x\) into the function \(f(x) = \sqrt[3]{x^3 - 5x}\). Let's find \(f(-x)\):\[f(-x) = \sqrt[3]{(-x)^3 - 5(-x)} = \sqrt[3]{-x^3 + 5x}.\]

Step 3: Compare f(-x) with f(x)

Let's compare \(f(-x)\) which is \(\sqrt[3]{-x^3 + 5x}\) with \(f(x) = \sqrt[3]{x^3 - 5x}\).Notice that:\[f(-x) = \sqrt[3]{-1(x^3 - 5x)} = -\sqrt[3]{x^3 - 5x}.\]So, \(f(-x) = -f(x)\), indicating the function is odd.

Key Concepts

Function Evaluation

Evaluating a function means plugging in values to determine how it behaves. In this context, we're evaluating the function at \(-x\) to see how the function changes.

• If you substitute \(-x\) into the function and get the original function back, the function is even.
• If you get the negative of the original function, the function is odd.

Key in this process is closely following the steps of substitution and algebraic simplification. Make sure each transformed term is correctly worked through, especially when negative signs are involved.

Cube Root Function

The cube root function is expressed as \(\sqrt[3]{x}\). It's vital to remember that cube roots apply to any real number, positive or negative.

• Cube roots maintain the sign of the original number — in contrast to square roots, they do not restrict themselves to positive numbers.
• This property is essential in evaluating functions for determining characteristics like oddness or evenness.
• In the example, \(\sqrt[3]{-8} = -2\), underlining the point that cube roots handle negative inputs differently compared to square roots.

Odd Function

An odd function has a specific symmetrical property concerning the origin. A function is classified as odd if it satisfies the condition \(f(-x) = -f(x)\).

• This symmetry implies that when plotted, the graph of an odd function reflects through the origin.
• Such functions typically involve terms with odd exponents, but the transformation and calculation confirm this.
• In the exercise example, substituting negative \(x\) into the function and simplifying shows \(f(-x) = -f(x)\), thus validating the function as odd.

Mathematical Comparison

Comparing \(f(-x)\) and \(f(x)\) involves algebraic manipulation to identify the function's nature. We check if \(f(-x)\) equals \(f(x)\) or its negative.

• This comparison involves ensuring each component of the function is correctly substituted and simplified.
• Look for direct equivalence or oppositional signs to label the function even, odd, or neither.
• Our example function, \(\sqrt[3]{x^3 - 5x}\), when evaluated at \(-x\) resulted in \(f(-x) = -f(x)\), cementing its classification as odd.

Remember, precision in algebraic steps is essential to ascertain the function's characteristics correctly.