Problem 85 - 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 \(x\). $$g(s)=4 s^{2 / 3}$$

Step-by-Step Solution

Answer

The function \(g(s)=4 s^{2 / 3}\) is even as it satisfies the conditions algebraically, graphically and numerically.

Step 1: Algebraic Evaluation

Substitute \(s\) with \(-s\): \(g(-s) = 4(-s)^{2/3} = 4s^{2/3}\). Since \(g(-s) = g(s)\), the function is even.

Step 2: Graphical Evaluation

Using a graphing utility, graph the function \(g(s)=4 s^{2 / 3}\). The graph is symmetric over the y-axis which indicates an even function.

Step 3: Numerical Evaluation

Using the table feature of the graphing utility, create a table comparing \(f(x)\) and \(f(-x)\) for several values of \(x\). If the result of \(f(x)\) is the same as that of \(f(-x)\), the function is even.

Key Concepts

Algebraic Evaluation

To determine if a function is even algebraically, we substitute the variable with its opposite and observe the outcome. For the function \(g(s) = 4s^{2/3}\), we substitute \(s\) with \(-s\).
After substitution, we have \(g(-s) = 4(-s)^{2/3}\). Due to the exponent \(2/3\) being positive and even in effect, the negative within the parentheses does not change the result. Therefore, \(g(-s) = 4s^{2/3}\).
This is equal to \(g(s)\), thus proving \(g(s)\) is even. An even function satisfies the equation \(g(-s) = g(s)\) for all values of \(s\). This approach can be used as a quick verification when determining the parity of a function.

Graphical Evaluation

Graphical evaluation involves visual inspection of the function's graph. For \(g(s)=4s^{2/3}\), using a graphing utility helps us see if the function is symmetric about the y-axis.
When you plot the function, observe the two halves of the graph. If they mirror each other across the y-axis, the function is even. This symmetry is crucial to identifying the nature of the function.

• Check for y-axis symmetry: The graph should look the same on both sides of the y-axis.
• Graphing tools can often highlight this symmetry automatically.

In our case, the graph is symmetric over the y-axis, which reinforces the conclusion that \(g(s)\) is even. This method offers a visual confirmation complementing algebraic findings.

Numerical Evaluation

Numeric evaluation uses a calculated approach to compare the function values. For \(g(s)=4s^{2/3}\), by choosing several values of \(x\), we calculate both \(g(x)\) and \(g(-x)\).
Examples could include \(x = 1, 2, 3,\) etc. We create a table with these results, such as:

• \(x = 1\), \(g(1) = 4(1)^{2/3} = 4\)
• \(-x = -1\), \(g(-1) = 4(-1)^{2/3} = 4\)
• \(x = 2\), \(g(2) = 4(2)^{2/3} = 2.52\) (approximately)
• \(-x = -2\), \(g(-2) = 4(-2)^{2/3} = 2.52\) (approximately)

In each instance, \(g(x)\) equals \(g(-x)\), confirming through numerical evidence that the function is even. This method provides a different perspective by practically verifying equal outcomes for mirrored inputs across the y-axis.