To determine if a function is even, odd, or neither, we examine the relationship between \(f(x)\) and \(f(-x)\).

Understanding Even and Odd Functions

Let's define what makes a function even or odd:

• Even Function: 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.
• Odd Function: A function \(f(x)\) is odd if \(f(-x) = -f(x)\) for all \(x\) in its domain. Odd functions are symmetric about the origin.
• Neither Even nor Odd: A function is neither even nor odd if it does not satisfy either of the above conditions.

Let's analyze each given function to see which category it falls into.

Analyzing Option 1: \(f(x) = x^2 - 1\)

Consider the function \(f(x) = x^2 - 1\). Let's find \(f(-x)\):

\(f(-x) = (-x)^2 - 1\)

Since \((-x)^2 = x^2\), we have:

\(f(-x) = x^2 - 1\)

Comparing \(f(-x)\) with \(f(x)\), we see that \(f(-x) = f(x)\).

Therefore, the function \(f(x) = x^2 - 1\) is an even function.

Analyzing Option 2: \(f(x) = x + \frac{3}{x}\)

Consider the function \(f(x) = x + \frac{3}{x}\). Let's find \(f(-x)\):

\(f(-x) = (-x) + \frac{3}{(-x)}\)

This simplifies to:

\(f(-x) = -x - \frac{3}{x}\)

We can factor out -1 from the expression:

\(f(-x) = -(x + \frac{3}{x})\)

Comparing \(f(-x)\) with \(f(x)\), we see that \(f(-x) = -f(x)\).

Therefore, the function \(f(x) = x + \frac{3}{x}\) is an odd function.

Analyzing Option 3: \(f(x) = |x|\)

Consider the function \(f(x) = |x|\). Let's find \(f(-x)\):

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

By the definition of absolute value, \(|-x| = |x|\).

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

Comparing \(f(-x)\) with \(f(x)\), we see that \(f(-x) = f(x)\).

Therefore, the function \(f(x) = |x|\) is an even function.

Analyzing Option 4: \(f(x) = x^2(x - 3)\)

Consider the function \(f(x) = x^2(x - 3)\). First, let's expand the function:

\(f(x) = x^3 - 3x^2\)

Now, let's find \(f(-x)\):

\(f(-x) = (-x)^3 - 3(-x)^2\)

Since \((-x)^3 = -x^3\) and \((-x)^2 = x^2\), we get:

\(f(-x) = -x^3 - 3x^2\)

Now, let's compare \(f(-x)\) with \(f(x)\) and \(-f(x)\).

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

\(-x^3 - 3x^2 = x^3 - 3x^2\)

This equation is true only if \(2x^3 = 0\), which means \(x = 0\). It is not true for all values of \(x\).

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

\(-f(x) = -(x^3 - 3x^2) = -x^3 + 3x^2\)

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

\(-x^3 - 3x^2 = -x^3 + 3x^2\)

This equation is true only if \(-3x^2 = 3x^2\), which means \(6x^2 = 0\), so \(x = 0\). It is not true for all values of \(x\).

Since \(f(-x)\) is neither equal to \(f(x)\) nor \(-f(x)\) for all \(x\) in its domain, the function \(f(x) = x^2(x - 3)\) is neither even nor odd.

Summary of Function Types

Based on our analysis, the function that is neither even nor odd is \(x^2(x - 3)\).

Revision Table: Function Properties

Additional Information: Graphing Even and Odd Functions

Understanding the graphs of even and odd functions can provide visual insight into their properties. An even function's graph looks the same if you reflect it across the y-axis. For example, the graph of \(y = x^2\) is symmetric about the y-axis. An odd function's graph looks the same if you rotate it 180 degrees about the origin. For example, the graph of \(y = x^3\) exhibits this rotational symmetry.

Functions that are neither even nor odd do not possess these specific types of symmetry about the y-axis or the origin. They might have other symmetries, or no symmetry at all.

To check if a function is neither even nor odd, it is sufficient to show that \(f(-x) \neq f(x)\) and \(f(-x) \neq -f(x)\) for at least one value of \(x\) (other than 0, as some conditions might hold only at \(x=0\)). However, the formal definition requires the condition to hold for all \(x\) in the domain, so checking the algebraic equality \(f(-x) = f(x)\) and \(f(-x) = -f(x)\) is the rigorous method.