Problem 83 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the function is even, odd, or neither (a) algebraically, (b) graphically by using a graphing utility to graph the function, and (c) numerically by using the table feature of the graphing utility to compare \(f(x)\) and \(f(-x)\) for several values of \(x\). $$f(x)=x \sqrt{1-x^{2}}$$

Step-by-Step Solution

Answer

The function \(f(x) = x\sqrt{1 - x^2}\) is neither even nor odd.

Step 1: Algebraic Analysis

Compute \(f(-x)\), if the result equals \(f(x)\) then the function is even, and if it equals \(-f(x)\), the function is odd. For \(f(-x) = -x\sqrt{1 - (-x)^2} = -x\sqrt{1-x^2}\). So \(f(x)\) does not equal \(f(-x)\) and also does not equal \(-f(x)\). Therefore, \(f(x)\) is neither even or odd.

Step 2: Graphical Analysis

By plotting the function \(f(x)\) in a graphing tool, one can see the graph is neither symmetrical about the y-axis nor about the origin. This confirms that the function is neither even or odd.

Step 3: Numerical Analysis

By substituting x with several values into \(f(x)\) and \(f(-x)\), one finds that \(f(x)\) does not equal \(f(-x)\) nor \(-f(x)\). This confirms through numerical analysis that the function is neither even or odd. A few examples: for \(x=0.5\), \(f(0.5) = -f(-0.5)\) and also for \(x=1\), \(f(1) \neq -f(-1)\).

Key Concepts

Algebraic Analysis

Algebraic analysis helps in understanding the nature of a function in terms of symmetry, specifically if it's even, odd, or neither. The primary step involves evaluating the function at

• both \(x\) and \(-x\).

The key rules are:

• If \(f(x) = f(-x)\), the function is even.
• If \(f(x) = -f(-x)\), the function is odd.

When analyzing \(f(x)=x \sqrt{1-x^{2}}\), we substitute \(-x\) into the function:

• \(f(-x) = -x \sqrt{1 - (-x)^2} = -x \sqrt{1-x^2}\).

The result neither equals \(f(x)\) nor \(-f(x)\). Hence, the function is neither even nor odd, as it aligns with neither of the symmetry conditions.

Graphical Analysis

Graphical analysis involves visually examining the graph of the function to determine its symmetry properties. Using a graphing utility allows us to observe:

• Symmetry around the y-axis, indicating an even function.
• Symmetry around the origin, indicating an odd function.

For the function \(f(x)=x \sqrt{1-x^{2}}\), plot it using a graphing calculator or software. You will notice that:

• The graph is not symmetrical around the y-axis.
• Nor is it symmetrical around the origin.

This visual lack of symmetry further confirms that the function is neither even nor odd. Graphical representation is a powerful tool in analyzing these properties where algebraic verification might seem abstract.

Numerical Analysis

Numerical analysis provides a concrete verification by checking values directly. This involves using a table of values to compare outputs:

• Calculate \(f(x)\) for various values of \(x\).
• Compute \(f(-x)\) for the same values.

For \(f(x)=x \sqrt{1-x^{2}}\), let's substitute some example values:

• When \(x = 0.5\), \(f(0.5)\) is not equal to \(-f(-0.5)\).
• When \(x = 1\), \(f(1)\) does not equal \(-f(-1)\).

These calculated outputs clearly indicate that \(f(x)\) neither equals \(f(-x)\) nor \(-f(-x)\). Observing this pattern across several data points aids in confirming the function's nature beyond algebraic and graphical methods.