Problem 46 - 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.) $$ f(x)=5 $$

Step-by-Step Solution

Answer

f(x) = 5 is a polynomial function.

Step 1: Understanding the Functions

First, we need to understand the types of functions: **Polynomials** are sums of terms \(a_n x^n\) where \(a_n\) is a coefficient and \(n\) is a non-negative integer. **Rational functions** are ratios of polynomials. **Exponential functions** have the form \(a^x\). **Piecewise linear functions** are defined by different linear expressions in different intervals. **None of these** indicates the function doesn't fit any of these forms.

Step 2: Analyzing f(x) = 5

The function \(f(x) = 5\) is a constant function. A constant can be thought of as \(5 = 5x^0\), which has the form of a polynomial with a degree of 0.

Step 3: Identifying the Type

Since \(f(x) = 5\) can be expressed as \(5x^0\), it fits the definition of a polynomial function. Therefore, \(f(x) = 5\) is a polynomial function of degree 0.

Key Concepts

Polynomial Functions

Polynomial functions are one of the most straightforward types of mathematical expressions. They consist of terms that are made up of a coefficient and a variable raised to a non-negative integer power. For example, the function \(f(x) = 4x^3 + 3x^2 - 2x + 1\) is a polynomial function because it sums multiple terms, each with a variable raised to a power like 3, 2, 1, or 0.

Constant functions, like \(f(x) = 5\), which can be expressed as \(5x^0\), are also considered polynomial functions, but with a degree of zero. Here, the degree indicates the highest power of the variable in the function which, in this case, is zero. This highlights the versatility of polynomial functions, as they can range from simple constants to complex expressions involving multiple terms and variables.

Characteristics of polynomial functions include:

• They are continuous and smooth, meaning no breaks or sharp angles.
• Their degree determines the number of solutions or roots they might have.
• The domain of polynomial functions is the set of all real numbers.

These attributes make polynomial functions widely applicable in mathematics and engineering.

Rational Functions

Rational functions can be a bit more complex than polynomial functions as they involve ratios of polynomials. A rational function is any function that can be expressed as the quotient of two polynomial functions. Essentially, it looks like \(f(x) = \frac{p(x)}{q(x)}\), where \(p(x)\) and \(q(x)\) are polynomials and \(q(x)\) is not zero.

Consider the function \(f(x) = \frac{2x^2 + 3}{x - 1}\). Here, the numerator is a polynomial of degree 2 and the denominator is a polynomial of degree 1.

Key attributes of rational functions include:

• The domain excludes any x values that make the denominator zero since dividing by zero is undefined.
• They can have vertical asymptotes at values that make the denominator zero.
• Rational functions can approach a horizontal asymptote, typically found by comparing the degrees of the numerator and the denominator.

Understanding rational functions is crucial, especially in calculus, as they often appear in rate and ratio problems.

Exponential Functions

Exponential functions provide a different perspective in mathematics by focusing on growth and decay. These functions are of the form \(f(x) = a^x\), where \(a\) is a positive constant called the base and \(x\) is the exponent. A classic example is \(f(x) = 2^x\).

These functions grow much faster than polynomial functions because the variable \(x\) is the exponent. The base \(a\) determines whether the function represents exponential growth (if \(a > 1\)) or exponential decay (if \(0 < a < 1\)).

Features of exponential functions include:

• The graph of an exponential function is always continuous and smooth.
• They have a constant ratio of change, meaning they multiply by the same factor over equal intervals.
• The domain is all real numbers, but the range is positive real numbers if the function is in its simplest form.

Exponential functions are particularly crucial in modeling real-world scenarios like population growth, radioactive decay, and interest calculations.