Problem 46 - A Graphical Approach to College Algebra

Question

Complete the table, assuming that \(g\) is an odd function. $$\begin{array}{c|c|c|c|c|c|c|c} x & -5 & -3 & -2 & 0 & 2 & 3 & 5 \\ \hline g(x) & 13 & & -5 & & & -1 & \end{array}$$

Step-by-Step Solution

Answer

Use odd function properties: \(g(5) = -13\), \(g(-3) = 1\), \(g(2) = 5\), \(g(0) = 0\).

Step 1: Understanding Odd Functions

An odd function is characterized by the property that for every point \(x\) in the function's domain, the function satisfies \(g(-x) = -g(x)\). This means that the function is symmetric with respect to the origin.

Step 2: Apply the Definition to Known Values

Given the value \(g(-5) = 13\), we use the property of odd functions to find \(g(5)\). Since \(g(-5) = -g(5)\), it follows that \(g(5) = -13\).

Step 3: Find the Value for \(x=3\)

We know \(g(3) = -1\). By the property of odd functions, \(g(-3) = -g(3)\), so \(g(-3) = -(-1) = 1\).

Step 4: Find the Value for \(x=-2\)

The given table has \(g(-2) = -5\). Applying the property \(g(-x) = -g(x)\), we find that \(g(2) = -(-5) = 5\).

Step 5: Determine the Value for \(x=0\)

For an odd function, \(g(0)\) is always zero, because \(g(0) = -g(0)\) implies \(g(0) = 0\).

Key Concepts

Function Symmetry

Function symmetry is a fascinating concept that helps us understand the behavior and appearance of functions on a graph. For an odd function, this symmetry is found around the origin of the graph. An important characteristic of odd functions relates to their algebraic definition: they satisfy the condition \(g(-x) = -g(x)\) for every value of \(x\) in their domain.

This symmetry means if you were to rotate the graph of an odd function by 180 degrees around the origin, it would look the same as it did before. This property becomes our guiding principle when analyzing or graphing odd functions, such as the one presented in the original problem where we completed a table using this symmetry rule.

• Visualize odd function symmetry through axis rotation.
• Remember: Symmetric over both quadrants about the origin.

Function Properties

Every function has its own set of defining properties that help to understand its behavior and characteristics. In the case of odd functions, they have a unique set of properties:

• Origin Symmetry: As mentioned earlier, they are symmetric about the origin, evidenced by the property \(g(-x) = -g(x)\).
• Function and Its Negative Counterpart: For every \(x\), the function value at positive \(x\) and its negative counterpart will be negatives of each other.
• Zero Value at the Origin: Because of the symmetry and the property \(g(0) = -g(0)\), the function value precisely at the origin must be zero—that’s why in the given problem, we concluded \(g(0) = 0\).

Understanding these properties allows us to predict and understand how the function behaves without needing to graph every point, thereby making calculations easier and patterns more visible.

Algebraic Functions

Algebraic functions include many important examples that can be described using a finite number of operations involving additions, subtractions, multiplications, and divisions. They can include odd functions, which represent certain types of algebraic functions where specific symmetries can simplify and predict values.

In the context of the odd function \(g\) from our example, recognizing it as an algebraic function helps us apply algebraic manipulation to find unknown values efficiently. By understanding the inherent properties and symmetries, such as knowing \(g(-y) = -g(y)\), we leverage simple arithmetic to deduce missing data without needing actual measurements or plots.

• Use algebraic operations to handle unknowns.
• Algebraic functions incorporate simple symmetries for simplification.
• Learn to use these properties to assist in solving function-based problems.