Analyzing Function Parity: \(f(x) = |x| - x^3\)

To determine if a function \(f(x)\) is even, odd, or neither, we evaluate \(f(-x)\) and compare it to the original function \(f(x)\) and its negative \(-f(x)\).

Definition of Even and Odd Functions

• A function \(f(x)\) is even if \(f(-x) = f(x)\) for all \(x\) in its domain. Even functions are symmetric about the y-axis.
• A function \(f(x)\) is odd if \(f(-x) = -f(x)\) for all \(x\) in its domain. Odd functions have rotational symmetry about the origin.
• If neither of these conditions holds true for all \(x\), the function is considered neither even nor odd.

Step-by-Step Analysis of \(f(x) = |x| - x^3\)

Let's consider the given function: \(f(x) = |x| - x^3\).

Now, we find \(f(-x)\) by substituting \(-x\) for \(x\) in the function:

\(f(-x) = |-x| - (-x)^3\)

We know that \(|-x| = |x|\) for any real number \(x\), and \((-x)^3 = (-x) \times (-x) \times (-x) = -x^3\).

Substituting these back into the expression for \(f(-x)\):

\(f(-x) = |x| - (-x^3)\)

\(f(-x) = |x| + x^3\)

Checking for Evenness

Is \(f(-x) = f(x)\)?

We have \(f(-x) = |x| + x^3\) and \(f(x) = |x| - x^3\).

For \(f(-x)\) to be equal to \(f(x)\), we would need \(|x| + x^3 = |x| - x^3\). This simplifies to \(x^3 = -x^3\), which means \(2x^3 = 0\). This is only true for \(x=0\). For any other value of \(x\) (e.g., \(x=1\), \(f(-1) = |-1| + (-1)^3 = 1 - 1 = 0\), but \(f(1) = |1| - (1)^3 = 1 - 1 = 0\). Let's try another value, e.g., \(x=2\). \(f(-2) = |-2| + (-2)^3 = 2 - 8 = -6\), \(f(2) = |2| - (2)^3 = 2 - 8 = -6\). Wait, this example seems to satisfy it. Let's recheck the algebra. \(f(-x) = |x| + x^3\). \(f(x) = |x| - x^3\). If \(f(-x) = f(x)\), then \(|x| + x^3 = |x| - x^3\). Subtracting \(|x|\) from both sides gives \(x^3 = -x^3\). Adding \(x^3\) to both sides gives \(2x^3 = 0\). This equation is only true when \(x=0\). For a function to be even, the condition \(f(-x) = f(x)\) must hold for *all* \(x\) in the domain, not just \(x=0\). For example, if \(x=1\), \(f(-1) = |-1| + (-1)^3 = 1 - 1 = 0\). \(f(1) = |1| - (1)^3 = 1 - 1 = 0\). Okay, \(f(-1) = f(1)\). Let's try \(x=-1\). \(f(-(-1)) = f(1) = 0\). \(f(-1) = 0\). Okay, this case still holds. Let's try a non-integer, say \(x=0.5\). \(f(-0.5) = |-0.5| + (-0.5)^3 = 0.5 - 0.125 = 0.375\). \(f(0.5) = |0.5| - (0.5)^3 = 0.5 - 0.125 = 0.375\). What is wrong? Let's re-evaluate \(f(x) = |x| - x^3\).

Let's try \(x = -2\). \(f(-2) = |-2| - (-2)^3 = 2 - (-8) = 2 + 8 = 10\).

Now let's check \(f(2)\). \(f(2) = |2| - (2)^3 = 2 - 8 = -6\).

Is \(f(-2) = f(2)\)? No, \(10 \neq -6\).

Therefore, \(f(-x) \neq f(x)\) for all \(x\). The function is not even.

Checking for Oddness

Is \(f(-x) = -f(x)\)?

We have \(f(-x) = |x| + x^3\).

First, let's find \(-f(x)\): \(-f(x) = -(|x| - x^3) = -|x| + x^3\).

For \(f(-x)\) to be equal to \(-f(x)\), we would need \(|x| + x^3 = -|x| + x^3\). This simplifies to \(|x| = -|x|\). This is only true when \(|x|=0\), which means \(x=0\). For any other value of \(x\), \(|x|\) is positive, while \(-|x|\) is negative, so they are not equal.

Using the example \(x=-2\):

\(f(-2) = 10\)

\(-f(2) = -(-6) = 6\)

Is \(f(-2) = -f(2)\)? No, \(10 \neq 6\).

Therefore, \(f(-x) \neq -f(x)\) for all \(x\). The function is not odd.

Conclusion on Function Parity

Since the function \(f(x) = |x| - x^3\) is neither even nor odd, it is classified as neither even nor odd.

Revision Table: Key Concepts

Additional Information on Function Types

Understanding function parity (even, odd, or neither) is important in calculus and other areas of mathematics. For instance, knowing if a function is even or odd can simplify definite integral calculations over symmetric intervals.

The function \(f(x) = |x| - x^3\) is a combination of an even function \(|x|\) and an odd function \(x^3\). In general, the sum or difference of an even function and an odd function results in a function that is neither even nor odd, unless one of the functions is identically zero.

Consider \(g(x) = |x|\) (even) and \(h(x) = x^3\) (odd). The function \(f(x) = g(x) - h(x)\).

\(f(-x) = g(-x) - h(-x)\)

Since \(g\) is even, \(g(-x) = g(x) = |x|\).

Since \(h\) is odd, \(h(-x) = -h(x) = -x^3\).

So, \(f(-x) = |x| - (-x^3) = |x| + x^3\).

We already established that \(|x| + x^3\) is generally not equal to \(|x| - x^3\) (f(x)) and generally not equal to \(-(|x| - x^3) = -|x| + x^3\) (-f(x)). This confirms that \(f(x) = |x| - x^3\) is neither even nor odd.