Problem 86 - 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 \(x\). $$f(s)=4 s^{3 / 5}$$

Step-by-Step Solution

Answer

The function \( f(s) = 4s^{3/5} \) is neither even nor odd when examined algebraically, graphically, and numerically.

Step 1: Algebraic Analysis

To determine whether the function is even or odd algebraically, substitute \( -s \) for \( s \) in the function: \( f(-s) = 4(-s)^{3/5} = -4s^{3/5} \). This is not equal to \( f(s) \) and is also not equal to \( -f(s) \). Thus, algebraically, the function is neither even nor odd.

Step 2: Graphical Analysis

Graphically, an even function is symmetric about the y-axis, and an odd function is symmetric about the origin. When \( f(s) = 4s^{3/5} \) is graphed using a graphing utility, it shows that the graph is not symmetric about the y-axis nor about the origin. Therefore, graphically, the function is also neither odd nor even.

Step 3: Numerical Analysis

To determine whether the function is even or odd numerically, select several values of \( s \), determine \( f(s) \) and \( f(-s) \), and compare. It will be found that for all selected values, \( f(s) \neq f(-s) \) and \( f(s) \neq -f(-s) \). Therefore, numerically, the function is also neither odd nor even.

Key Concepts

Algebraic Analysis

In algebraic analysis, we determine whether a function is even, odd, or neither by examining its algebraic form. A function is considered an even function if, for all values of the variable, substituting the negation of the variable leads back to the original function. This is described with the rule:

• A function is even if: \( f(-x) = f(x) \).

An odd function, on the other hand, is defined such that substituting the negative of the variable results in the negation of the entire function:

• A function is odd if: \( f(-x) = -f(x) \).

For the function \(f(s) = 4s^{3/5}\), substituting \(-s\) in place of \(s\) yields: \[ f(-s) = 4(-s)^{3/5} = -4s^{3/5} \]. When compared to \(f(s) = 4s^{3/5}\), it doesn't satisfy either condition for evenness or oddness, thus algebraically, the function is neither even nor odd.

Graphical Analysis

Graphical analysis involves looking at the graph of the function to understand its properties. An even function will have symmetry about the y-axis. This means if you fold the graph along the y-axis, both halves will perfectly overlap. An odd function, on the other hand, has symmetry about the origin. This type of symmetry means if you rotate the graph 180 degrees around the origin, it remains unchanged.Using a graphing utility to plot the given function \( f(s) = 4s^{3/5} \), observe the behavior of the graph. You'll find that it neither exhibits symmetry across the y-axis nor demonstrates rotational symmetry through the origin. Therefore, based on graphical analysis, the function is determined to be neither even nor odd.

Numerical Analysis

Numerical analysis verifies the algebraic and graphical analysis by examining specific function values. To determine evenness or oddness numerically, evaluate the function at several points for \(s\) and their negatives \(-s\). Compare \(f(s)\) with \(f(-s)\) and \(-f(s)\) at these points. For the function \(f(s) = 4s^{3/5}\), you can generate pairs such as \((s=1, s=-1)\) and \((s=2, s=-2)\). Calculating these, you observe that for \(s = 1\), \(f(1) = 4 \times 1^{3/5} = 4\), and \(f(-1) = -4\), which is close yet still unequal to \(-4 \). Similarly, checking more values, such as \(s = 2\) and \(-2\), will confirm that neither condition for even \(f(s) = f(-s)\) nor odd \(f(s) = -f(-s)\) is met.Thus, numerically as well, the function does not conform to being even or odd.