Problem 41 - College Algebra and Calculus: An Applied Approach

Question

Sketch the graph of the function and determine whether the function is even, odd, or neither. \(g(s)=\frac{s^{3}}{4}\)

Step-by-Step Solution

Answer

The function \(g(s)=\frac{s^{3}}{4}\) is odd. The graph of the function shows symmetry about the origin, indicating its odd parity.

Step 1: Determine the parity of the function

The function given is \(g(s)=\frac{s^{3}}{4}\). The power of the variable 's' is 3, which is an odd number. Thus the function itself is odd.

Step 2: Graph the function

To sketch the graph of the function, plot a few points by substituting some values for 's' in the function: for example, s = -2, -1, 0, 1, 2. Then draw a smooth curve through these points. An odd function such as \(g(s)=\frac{s^{3}}{4}\) will show symmetry about the origin, i.e. it will be mirrored in the second and fourth quadrants.

Step 3: Verify the result

A function is odd when \(g(-s) = -g(s)\) for all 's' in the domain. To make sure that this function is odd, substitute \(-s\) in place of \(s\) in the original function: this gives \(\frac{(-s)^{3}}{4} = -g(s)\). Thus, \(g(s)\) is indeed an odd function.

Key Concepts

Odd Functions

Odd functions are an interesting type of function that exhibit unique properties. A function is considered odd if for every input value \(s\), the output with its sign switched, \(-g(s)\), is equal to the function evaluated at \(-s\): \(g(-s) = -g(s)\). This means that when you replace \(s\) with \(-s\) in the function, the value should be the opposite of the original function value.
- Odd functions are related to the power of the variable in the expression. - For functions like \(g(s) = \frac{s^3}{4}\), where the highest power is an odd number, the function tends to be odd.
Odd functions have a special symmetry, specifically rotational symmetry about the origin, which we'll explore below.

Function Graphing

Graphing functions helps us visualize the behavior of the function. When sketching the graph of \(g(s)=\frac{s^3}{4}\), you can follow these simple steps:

• Start by selecting a few points to plot. A good strategy is to use a few positive and negative values, along with zero, like \(s = -2, -1, 0, 1, 2\).
• Calculate the corresponding \(g(s)\) values for each chosen \(s\). For instance, if \(s = 2\), then \(g(2) = \frac{2^3}{4} = 2\).
• Plot these points on the graph.
• Join the points smoothly to form the curve.

In this graph, you'll notice that the curve passes through the origin and appears mirrored about it, confirming that the function displays origin symmetry, typical for odd functions.

Symmetry in Functions

Symmetry in functions can provide quick clues about the nature of the function. There are two common types of symmetry to watch for: symmetry about the y-axis and symmetry about the origin.
- **Y-axis symmetry** is found in even functions, where \(g(s) = g(-s)\) for all \(s\). This means the left and right sides of the graph are mirror images.
- **Origin symmetry**, on the other hand, is found in odd functions, like our function \(g(s)=\frac{s^3}{4}\). This symmetry means that rotating the graph by 180 degrees about the origin yields the same graph. For odd functions, every point \((s, g(s))\) has a corresponding point \((-s, -g(s))\) on the curve.
Recognizing these symmetries can save time in graphing by providing immediate information about how the graph looks without plotting numerous points. This symmetry is a defining characteristic when quickly determining whether the function is odd or even.