Problem 39 - 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.) $$ f(x)=\frac{1}{x+2} $$

Step-by-Step Solution

Answer

Rational function.

Step 1: Identify the Basic Form of the Function

The given function is \( f(x) = \frac{1}{x+2} \). Notice it is written as a fraction where the numerator is constant 1 and the denominator is \( x+2 \), which contains a variable.

Step 2: Determine if it is a Polynomial

A polynomial function has terms that are non-negative integer powers of \( x \). Since \( f(x) \) has a denominator that involves \( x \), it does not fit this definition of a polynomial.

Step 3: Determine if it is a Rational Function

A rational function is a ratio of two polynomials, where the numerator and the denominator are both polynomials. In \( f(x) = \frac{1}{x+2} \), the numerator (1) and the denominator \( (x+2) \) are polynomials, thus \( f(x) \) is a rational function.

Step 4: Determine if it is an Exponential Function

An exponential function has the form \( a^x \) for a constant base \(a\). Since there is no exponent with base as a variable, \( f(x) \) is not an exponential function.

Step 5: Determine if it is a Piecewise Linear Function

A piecewise linear function is defined by multiple linear segments over different intervals of the input variable. Since \( f(x) = \frac{1}{x+2} \) is continuous and does not have defined intervals with distinct linear equations, it is not a piecewise linear function.

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions involving sums of terms where each term includes a variable raised to a non-negative integer exponent. For example, a polynomial function like \( f(x) = 3x^2 + 2x + 1 \) consists of terms with exponents 2, 1, and 0.Some key characteristics of polynomial functions include:

• They are continuous and smooth curves without sharp turns or breaks.
• The highest power of the variable (degree) determines the behaviour of the function as the variable goes to infinity.
• They can be as simple as a constant (e.g., \( f(x) = 5 \)) or involve multiple terms with different powers of \( x \).

Polynomial functions are not limited by the values of the variables, unlike rational functions that have restrictions in the denominator.

Exponential Functions

Exponential functions have a fundamental structure where a constant base is raised to a variable exponent, typically expressed as \( f(x) = a^x \), where the base \( a \) is a constant and \( x \) is the variable. Unlike polynomial functions, exponential functions involve the variable as an exponent.Important features of exponential functions include:

• The graph of an exponential function showcases rapid growth or decay. This depends on whether the base is greater than or less than 1.
• Exponential functions always yield positive results because exponents determine the value's direction of growth (above or below 1), rather than magnitude.
• They exhibit a constant rate of relative change, which is a unique feature compared to other types of functions.

Exponential functions are vastly used in modeling natural phenomena like population growth, radioactive decay, and compound interest, due to their rapid and consistent growth patterns.

Piecewise Linear Functions

Piecewise linear functions are defined using multiple linear expressions, each applicable over different intervals of the function's domain. They can accurately model situations with different behaviours in separate sections, such as tax brackets or shipping rates.Characteristics of piecewise linear functions include:

• Each segment is linear, meaning they have a consistent slope or rate of change.
• The overall function is 'piecewise,' suggesting different rules or formulas over specified intervals of \( x \).
• The change in intervals may involve jumps or points of discontinuity, making them different from continuous polynomial or rational functions.

Understanding the application and definition of intervals is crucial when dealing with piecewise linear functions, as real-world situations often involve changing conditions that such functions can model effectively.