Problem 58 - A Graphical Approach to College Algebra

Question

Each function is either even or odd. Use \(f(-x)\) to state which situation applies. $$f(x)=8$$

Step-by-Step Solution

Answer

The function is even.

Step 1: Understanding Even and Odd Functions

A function is called even if it satisfies the condition \( f(-x) = f(x) \) for all inputs \( x \). A function is odd if it satisfies \( f(-x) = -f(x) \). To determine if a function is even or odd, we substitute \(-x\) into the function and compare it to the original function.

Step 2: Evaluate f(-x)

Given the function \( f(x) = 8 \), we substitute \(-x\) into the function. Since the function is constant, \( f(-x) = 8 \).

Step 3: Determine the Function Type

Compare \( f(-x) \) and \( f(x) \). We have \( f(-x) = 8\) and \( f(x) = 8\). Since \( f(-x) = f(x) \), the function is even.

Key Concepts

Constant Function

A constant function is a special type of function where the output value is the same, regardless of the input. In the simplest terms, no matter what value you plug in for the variable, the output remains constant, hence the name "constant function." For example, the function \( f(x) = 8 \) is a constant function. This is because, whether you input 5, -3, 0, or any other number, the output will always be 8.

Constant functions have some interesting characteristics:

• The graph of a constant function is a horizontal line. In our example, \( f(x) = 8 \) would be a straight line parallel to the x-axis, intersecting the y-axis at (0,8).
• Constant functions are both even and odd. However, in our context of binary classification (either even or odd), the convention is to identify them as even because \( f(-x) = f(x) \).
• They have zero slope, meaning there is no change as x changes.

This lack of change (or uniformity) makes analyzing constant functions straightforward and an essential part of understanding more complex functions.

Function Properties

Functions have various properties that classify and describe their behavior. Knowing these properties helps us understand different function types better. When analyzing a function, these are some key properties to consider:

• Even and Odd Functions: As mentioned, even functions satisfy \( f(-x) = f(x) \) and are symmetric about the y-axis. Odd functions satisfy \( f(-x) = -f(x) \) and are symmetric about the origin.
• Domain and Range: The domain is the set of all possible inputs (x-values) and the range is the set of all possible outputs (y-values).
• Continuity: This describes whether a function has any breaks, holes, or gaps when graphed. Constant functions are continuous, as they are unbroken lines.
• Slope and Monotonicity: While constant functions have a slope of 0 (flat line), other functions can have varying slopes, indicating whether they are increasing or decreasing.

These function properties allow us to categorize functions and forecast their behavior under different conditions. Understanding these concepts is fundamental to algebra and calculus.

Algebraic Functions

Algebraic functions are functions built from algebraic operations: addition, subtraction, multiplication, division, and taking roots. These functions can include polynomials, rational functions, and radical functions.

• Polynomials: These are sums of terms consisting of variables raised to whole number powers, like \( f(x) = x^2 + 2x + 1 \).
• Rational Functions: These are ratios of two polynomials, such as \( f(x) = \frac{x+1}{x-1} \). Rational functions can be more complex than simple polynomials due to their division aspect, which can introduce asymptotes.
• Radical Functions: These involve roots, like \( f(x) = \sqrt{x} \), and often bring about domain restrictions because certain inputs might not result in real numbers.

Constant functions can be considered the simplest algebraic functions because they don't change; they involve multiplication by zero and addition, making them a special case of polynomial functions. Algebraic functions are foundational in algebra as they describe many real-world phenomena and can be manipulated to solve complex equations.