Problem 2 - 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. $$ \begin{array}{|c|c|c|} \hline x & -3 & 3 \\ \hline f(x) & & -5 \\ \hline g(x) & & 15 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

f(-3) = -5, g(-3) = -15.

Step 1: Understanding Even and Odd Functions

An even function, such as \( f(x) \), satisfies the property \( f(-x) = f(x) \). This means that the output for \( x \) and \( -x \) is the same. An odd function, such as \( g(x) \), satisfies the property \( g(-x) = -g(x) \), meaning the output for \( -x \) is the opposite of the output for \( x \).

Step 2: Complete Table for f(x) when x = -3

Since \( f(x) \) is even, we have \( f(-3) = f(3) \). From the table, \( f(3) = -5 \). Thus, \( f(-3) = -5 \).

Step 3: Complete Table for g(x) when x = -3

Since \( g(x) \) is odd, we have \( g(-3) = -g(3) \). Given in the table \( g(3) = 15 \), so \( g(-3) = -15 \).

Step 4: Finalize the Table

Fill in the missing values in the table using the conclusions from steps 2 and 3: \[\begin{array}{|c|c|c|} \hline x & -3 & 3 \ \hline f(x) & -5 & -5 \ \hline g(x) & -15 & 15 \ \hline \end{array} \]

Key Concepts

Function Properties

When we talk about functions in mathematics, understanding their properties is key to solving many types of problems. In this context, property refers to the unique characteristics and behaviors of functions that help us predict their behavior in different scenarios.

Consider the case of even and odd functions:

• An **even function** has the property that its value remains unchanged when the input sign is reversed. Mathematically, this is represented as \( f(-x) = f(x) \).
• An **odd function**, on the other hand, shows antisymmetry. This means if we reverse the sign of the input, the sign of the output is also reversed. This is expressed by \( g(-x) = -g(x) \).

These properties of functions are not just theoretical. They form the foundation for understanding symmetry, aiding us in predicting and visualizing function behavior.

Symmetry in Functions

Symmetry in functions is an important concept that makes understanding and graphing functions much simpler. Functions can exhibit different types of symmetry, namely, even (mirror symmetry across the y-axis) and odd (rotational symmetry around the origin).

**Even Function Symmetry:**

• For even functions, symmetry means the graph will look the same to the left and right of the y-axis.
• Mathematically, if you plug in \(-x\) into an even function and get exactly the same result as plugging in \(x\), that function is exhibiting symmetry and is even.

**Odd Function Symmetry:**

• Odd function symmetry involves a sort of rotational symmetry around the origin. If you rotate the graph 180 degrees around the origin, it looks the same.
• This is verified mathematically when \( g(-x) = -g(x) \), indicating that the shape flips around the origin when you change the sign of the input.

Symmetry is a powerful tool that simplifies analyzing and sketching functions.

Algebraic Functions

Algebraic functions encompass a wide variety of equations derived from algebraic operations such as addition, subtraction, multiplication, division, and root extractions. These functions are fundamental in mathematics because they describe a range of phenomena and simplify computations.

**Types of Algebraic Functions:**

• Polynomial Functions: These include functions like \( f(x) = x^2 \) or \( g(x) = x^3 \), building blocks of more complex algebraic expressions.
• Rational Functions: These are ratios of polynomials, such as \( f(x) = \frac{x^2 + 1}{x + 3} \).

Each type of algebraic function presents its own unique symmetries and properties. For instance, quadratics (like \( x^2 \)) are even, while cubics (like \( x^3 \)) are odd.

Understanding these types helps grasp concepts like symmetry and transformation in mathematical analysis, making algebraic functions a fundamental topic in mathematics.