Problem 84 - 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 \(x\). $$f(x)=x \sqrt{x+5}$$

Step-by-Step Solution

Answer

The function \(f(x)=x \sqrt{x+5}\) is neither even nor odd.

Step 1: Algebraic Analysis

Substitute \(x\) with \(-x\) in the equation: \(f(-x)=-x \sqrt{-x+5}\). This does not equal \(f(x)\), nor does it equal \(-f(x)\), which means the function is neither even nor odd algebraically.

Step 2: Graphical Analysis

By graphing the function \(f(x)=x \sqrt{x+5}\), we can see that the graph doesn't display symmetry with respect to the y-axis or the origin. Therefore, the function is neither even nor odd graphically.

Step 3: Numerical Analysis

Choose several values for \(x\) and calculate \(f(x)\) and \(f(-x)\). Comparing these values shows that \(f(x) \neq f(-x)\) and \(f(x) \neq -f(-x)\), supporting the conclusion that the function is neither even nor odd.

Key Concepts

Graphical Analysis

Understanding whether a function is even or odd through graphical analysis involves observing the symmetry of its graph.
The function's graph is plotted, and key symmetries are examined:

• An even function has symmetry about the y-axis.
• An odd function has rotational symmetry around the origin.

For the function \[f(x) = x \sqrt{x+5}\], examining the graph shows neither y-axis symmetry nor origin symmetry.
This confirms graphically that the function is neither even nor odd.

Numerical Analysis

Numerical analysis involves evaluating the function at multiple points to understand its behavior better.
This step allows us to compare the values of \(f(x)\) and \(f(-x)\):

• If \(f(x) = f(-x)\) for all \(x\), the function is even.
• If \(f(x) = -f(-x)\) for all \(x\), the function is odd.
• If neither condition is satisfied, the function is neither.

For example, testing a few points on the function \(f(x) = x \sqrt{x+5}\), such as \(x=0\), \(x=1\), and \(x=-1\), confirms that neither condition fits. Thus, numerically, the function is neither even nor odd.

Algebraic Analysis

To determine the nature of the function algebraically, we substitute \(x\) with \(-x\) in \(f(x)\):
\[ f(-x) = -x \sqrt{-x+5} \]
Comparing, neither \(f(-x) = f(x)\) nor \(f(-x) = -f(x)\) holds.
This result indicates that the function is neither even (no y-axis symmetry) nor odd (no origin symmetry) algebraically.
Algebraic analysis confirms why neither type of symmetry exists for this specific function.