Problem 38 - Brief Applied Calculus

Question

Identify each function as a polynomial, a rational function, an exponential function, a piecewise linear function, or none of these. (Do not graph them; just identify their types.) $$ f(x)=\left\\{\begin{array}{ll} 3 x-1 & \text { if } x \geq 2 \\ 1-x & \text { if } x<2 \end{array}\right. $$

Step-by-Step Solution

Answer

The function is a piecewise linear function.

Step 1: Recognize the structure

The function given is defined with different expressions for different intervals of the variable. This is a hallmark of a piecewise function. Each piece is defined by specific rules based on the value of \( x \).

Step 2: Analyze each piece

Examine each part of the piecewise function: 1. For \( x \geq 2 \), the expression is \( f(x) = 3x - 1 \). This is a linear function because it is in the form \( ax + b \), where \( a = 3 \) and \( b = -1 \).2. For \( x < 2 \), the expression is \( f(x) = 1 - x \), also a linear function since it's in the same linear form with \( a = -1 \) and \( b = 1 \).

Step 3: Identify the overall function type

Since both segments of the function are linear, and the function changes its definition based on the value of \( x \), this type of function is known as a piecewise linear function.

Key Concepts

Understanding Linear Functions

A linear function is one of the simplest forms of functions in mathematics. Its general form is expressed as \( f(x) = ax + b \), where \( a \) and \( b \) are constants. This means that the relationship between \( x \) and \( f(x) \) is a straight line when graphed. Linear functions are important because they form the basis for more complex mathematical concepts.

Key characteristics of linear functions include:

• The graph is a straight line.
• The rate of change or slope is constant, represented by \( a \).
• They have no exponents higher than 1 on the variable \( x \).
• The y-intercept is \( b \), where the line crosses the y-axis.

Linear functions are foundational in calculus and are widely used in various fields such as economics, physics, and engineering. They are valuable for predicting outcomes from simple input values.

Exploring Function Types

Functions come in various types, each with unique properties and applications. Identifying the type of function is crucial for solving mathematical problems effectively. Here is a quick overview of some common function types:

• Polynomial Functions: These include terms with powers of \( x \) like \( ax^n + bx^{n-1} + \ldots \). They can have multiple turning points.
• Rational Functions: These are ratios of two polynomials \( \frac{p(x)}{q(x)} \), and can have vertical asymptotes.
• Exponential Functions: These have a constant base raised to a variable exponent \( a^x \), showcasing rapid growth or decay.
• Piecewise Functions: These consist of multiple sub-functions, each applied to certain intervals of the domain.

Understanding these basic types helps in categorizing unfamiliar functions, predicting behavior, and applying the right methods for analysis.

Delving into Piecewise Functions

Piecewise functions are a fascinating type of function where different expressions are defined over different parts of the domain.

This structure allows the function to behave differently under various conditions. In a piecewise function like the one presented in the exercise, you will see multiple rules, each applied to specific intervals of \( x \).

• The segments do not necessarily have to be linear; they can be nonlinear as well, though in our exercise, they're linear.
• They allow complex modeling of a situation where conditions change suddenly. Think about tax brackets in economics or step-by-step engineering processes.
• Graphically, they can form a series of different shapes connected together.

Piecewise functions are incredibly useful in real-world scenarios because they can model scenarios that aren't constant and are subject to change. Understanding how to identify and work with these functions is essential for solving complex, variable-based problems efficiently.