Problem 38 - 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 \(f(x)\) is a piecewise linear function.

Step 1: Analyze the Structure of the Function

We have a function defined by different expressions for different intervals of the variable \(x\). This indicates that it is a type of function that changes its expression based on the input value of \(x\).

Step 2: Understand the Definition of Piecewise Functions

A piecewise function is a function that is defined by multiple sub-functions, each of which applies to a certain interval of the domain. In this case, \(f(x)\) is defined by two linear expressions, \(3x-1\) and \(1-x\), each over different intervals.

Step 3: Recognize the Linear Expressions

Both expressions \(3x - 1\) and \(1 - x\) are linear functions because they have the form \(ax + b\), where \(a\) and \(b\) are constants. Linear functions are a subset of polynomial functions of degree 1.

Step 4: Classify the Function Type

Since \(f(x)\) is composed of different linear expressions defined over distinct intervals, and each part is a polynomial of degree 1, \(f(x)\) is classified as a piecewise linear function.

Key Concepts

Function Classification

Functions can be categorized into various types based on their characteristics and the forms they take. Understanding these classifications helps in identifying and analyzing different mathematical functions. There are several common categories:

• Polynomial Functions: Functions that can be expressed as the sum of powers of a variable, like \(ax^n + bx^{n-1} + \ldots + k\), where \(n\) is a non-negative integer.
• Rational Functions: Functions that are ratios of two polynomial functions.
• Exponential Functions: Functions where a constant is raised to the power of a variable, such as \(a^x\).
• Piecewise Functions: Functions that have different expressions over different intervals of their domain.
• None of the Above: Some functions do not fit into any typical categories listed above.

In particular, the function in the original exercise is a piecewise linear function because it is defined by different linear expressions on different intervals. This classification is central to solving the exercise correctly.

Polynomial Functions

Polynomial functions form a significant group in function classification. A polynomial function is composed of variables raised to whole number powers, and these terms are summed together. They have a general form \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\), where \(n\) is a non-negative integer, and all coefficients \(a_n, a_{n-1},\) etc., are constants. Polynomial functions are classified further by their degree:

• Degree 0: Constant functions, such as \(f(x) = c\).
• Degree 1: Linear functions, like the ones seen in the exercise \(3x - 1\) and \(1 - x\).
• Degree 2: Quadratic functions, such as \(ax^2 + bx + c\).

In our exercise, both expressions used in the piecewise definition of the function are polynomials of degree 1, classifying them as linear functions, which we will delve into next.

Linear Functions

Linear functions are a simple yet powerful type of polynomial function that are used extensively in mathematics due to their straightforward nature and applications. They take the form \(f(x) = ax + b\), where \(a\) and \(b\) are constants.Key characteristics include:

• Slope \(a\): Represents the rate of change, or how steep the function is.
• Y-intercept \(b\): The point where the line crosses the y-axis.
• Graph: A straight line in the coordinate plane.

Both expressions, \(3x - 1\) and \(1 - x\), within the piecewise function are linear. This makes them easy to handle and predict, especially when analyzing piecewise linear functions.

Mathematical Expressions

In mathematics, expressions are combinations of numbers, variables, and operators that form meaningful computations. Understanding how to read and use expressions is crucial in algebra and calculus.Key elements of mathematical expressions include:

• Terms: Parts of an expression separated by addition or subtraction.
• Operators: Symbols like \(+, -, \times, \div\) that indicate operations.
• Variables: Symbols that represent unspecified numbers, typically \(x, y, z\).

Expressions like \(3x - 1\) and \(1 - x\) are examples used in the piecewise function from the exercise. Each expression is crafted from constants and variables under specific arithmetic operations, which determine the behavior and category of the function they constitute.