Problem 11 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 3-12: Determine whether \(f\) is even, odd, or neither even nor odd. $$ f(x)=\sqrt[3]{x^{3}-x} $$

Step-by-Step Solution

Answer

The function is odd.

Step 1: Define an Even Function

A function is even if for every \( x \) in its domain, \( f(-x) = f(x) \). This means that the graph of the function is symmetric with respect to the y-axis.

Step 2: Define an Odd Function

A function is odd if for every \( x \) in its domain, \( f(-x) = -f(x) \). This means that the graph of the function is symmetric with respect to the origin.

Step 3: Substitute \(-x\) into \(f(x)\)

We need to find \( f(-x) \) by substituting \(-x\) for \(x\) in the function \( f(x) = \sqrt[3]{x^3 - x} \). This gives us \( f(-x) = \sqrt[3]{(-x)^3 - (-x)} = \sqrt[3]{-x^3 + x} \).

Step 4: Test the Even Condition

Check whether \( f(-x) = f(x) \). We have \( f(x) = \sqrt[3]{x^3 - x} \) and \( f(-x) = \sqrt[3]{-x^3 + x} \). Clearly, \( \sqrt[3]{-x^3 + x} eq \sqrt[3]{x^3 - x} \), so the function is not even.

Step 5: Test the Odd Condition

Check whether \( f(-x) = -f(x) \). Since \( f(-x) = \sqrt[3]{-x^3 + x} \) and \( -f(x) = -\sqrt[3]{x^3 - x} = \sqrt[3]{-x^3 + x} \), we find that indeed \( f(-x) = -f(x) \). This confirms that the function is odd.

Key Concepts

Even Functions

Even functions exhibit a unique property where their behavior is mirrored across the y-axis. When a function is even, it means that if you were to fold its graph along the y-axis, the two halves would perfectly align. This important property can be mathematically validated by checking if the function satisfies \( f(-x) = f(x) \) for every \( x \) in its domain.

• If you can substitute \( -x \) into the function and achieve the same result as the original \( x \), then the function is even.
• This symmetry results in an equally spaced mirror image of the points on the graph about the y-axis.

It's also important to recognize that not all functions are even. For example, the function \(f(x) = x^2\) is even because \(f(-x) = f(x)\), hence it reflects evenly across the y-axis.

Function Symmetry

Function symmetry refers to certain consistent patterns in the graph of a function. Symmetry can occur in different forms, such as axis symmetry and point symmetry, each providing insightful properties for understanding a function's behavior.

• Axis Symmetry: This typically refers to symmetry along the y-axis, often seen in even functions. As we've discussed, even functions like \( f(x) = x^2 \) do not change when \( x \) is replaced by \(-x\). This indicates y-axis symmetry.
• Point Symmetry: Another type of symmetry is about the origin, associated with odd functions. A function exhibits origin symmetry when, after rotating by 180 degrees around the origin, the graph appears unchanged.

Consider odd functions like \( f(x) = x^3 \), which exhibit point symmetry. For such functions, \( f(-x) = -f(x) \) for each \( x \) in the domain.

Function Domains

Understanding the domain of a function is crucial as it tells us the set of all possible input values \( x \) for which the function is defined.

• The domain determines where the function can be evaluated and consequently affects the determination of symmetry properties.
• For example, if a function is defined only for non-negative values of \( x \), it can't exhibit the properties of an odd function, which require symmetry around the origin.

When analyzing more complex functions, such as \( f(x) = \sqrt[3]{x^3 - x} \), it's important to note that they often have domains defined on the entire set of real numbers due to the cubic root operation, allowing the function to maintain its properties of oddness or evenness across the full spectrum of \( x \). This specific function, due to the compatibility of cube roots with all real numbers, allows us to explore its attributes of symmetry fully.