Problem 1 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-2: Suppose \(f\) is an even function and \(g\) is an odd function. Complete the table, if possible. $$ \begin{array}{|l|r|r|} \hline x & -2 & 2 \\ \hline f(x) & & 7 \\ \hline g(x) & & -6 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

\( f(-2) = 7 \) and \( g(-2) = 6 \).

Step 1: Understand the properties of even and odd functions

An even function satisfies the condition \( f(-x) = f(x) \) for all \( x \). An odd function satisfies \( g(-x) = -g(x) \) for all \( x \). This information will help us fill in the table based on these properties.

Step 2: Fill the table for the even function \(f(x)\)

Given that \( f(x) \) is even and \( f(2) = 7 \), we use the property \( f(-x) = f(x) \). Therefore, \( f(-2) = f(2) = 7 \). Fill in \( f(-2) \) as 7 in the table.

Step 3: Fill the table for the odd function \(g(x)\)

Given that \( g(x) \) is odd and \( g(2) = -6 \), we use the property \( g(-x) = -g(x) \). Thus, \( g(-2) = -g(2) = -(-6) = 6 \). Fill in \( g(-2) \) as 6 in the table.

Key Concepts

Function Properties

Understanding the properties of even and odd functions is essential for solving many mathematical problems. An **even function** is defined by the symmetry it exhibits with respect to the y-axis, meaning that it satisfies the condition \( f(-x) = f(x) \) for any value of \( x \). This tells us that if you have a known value for \( f(x) \) at some point, then the value of \( f(-x) \) will be the same.On the other hand, an **odd function** exhibits rotational symmetry around the origin. This means it satisfies the condition \( g(-x) = -g(x) \) for all \( x \). Therefore, knowing the function value for a positive \( x \), you can easily determine the function value at \( -x \) by simply changing its sign. These fundamental properties of even and odd functions allow us to determine unknown function values from known ones.

Table Completion

Completing a table for even and odd functions involves applying their respective properties. Let's break it down with the table from our original exercise:

• We know \( f(x) \) is an even function, and the value when \( x = 2 \) is 7. Applying the even function property \( f(-x) = f(x) \), it follows that \( f(-2) = 7 \).
• Similarly, for the odd function \( g(x) \), we know that when \( x = 2 \), \( g(x) = -6 \). Using the odd function property \( g(-x) = -g(x) \), this results in \( g(-2) = 6 \).

By using these fundamental properties, filling in the missing table entries becomes straightforward. Understanding this application helps build intuition for function behaviors without complex calculations.

Problem Solving Skills

Solving problems involving functions can be greatly enhanced by a solid grasp of properties like evenness and oddness. Here are a few strategies to employ when tackling these kinds of problems:

• First, always identify whether a function is even, odd, or neither. This will guide the type of symmetry you can expect.
• Use symmetries to your advantage: Once you know a key property (\( f(-x) = f(x) \) or \( g(-x) = -g(x) \)), filling in the gaps becomes systematic.
• Work through each point methodically by applying the properties as seen in the example with \( x = 2 \) and \( x = -2 \).
• Visualizing the behavior of these functions graphically can also support understanding, showing how even functions mirror across the y-axis and odd functions reflect through the origin.

Developing these problem-solving skills helps make sense of function-related challenges, turning complex-looking tasks into manageable steps.

Step by Step Solution

The exercise solution is a great example of breaking down problems into manageable steps. Here's a streamlined look at how you can tackle similar functions:**Step 1**: Grasp the function properties. Recognize whether a function is even or odd to understand what symmetry it follows.**Step 2**: Apply these properties to known values in the function. For an even function, ensure \( f(-2) = f(2) \). For the odd function, calculate \( g(-2) \) from \( g(2) \) by applying the property \( g(-x) = -g(x) \).**Step 3**: Fill in the table with these values. This methodical filling helps avoid errors and ensures all function properties are honored.This step-by-step approach isn't just about reaching the answer, but about learning the process and understanding why each step is necessary. Such methods reinforce strong mathematical reasoning and skills applicable to a broad range of problems.