Problem 11 - College Algebra

Question

Identify the function as a power function, a polynomial function, or neither. $$f(x)=3^{x+1}$$

Step-by-Step Solution

Answer

The function is neither a power nor a polynomial function.

Step 1: Understand a Power Function

A power function is defined as a function of the form \(f(x) = kx^n\), where \(k\) is a constant and \(n\) is a non-negative real number. Examples include \(f(x) = 2x^3\) and \(f(x) = 5x^{-1}\). Identify whether the given function follows this form.

Step 2: Understand a Polynomial Function

A polynomial function is defined as a function of the form \(f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0\) where \(a_n, a_{n-1}, ..., a_0\) are constants and \(n\) is a non-negative integer. Identify if the given function fits this form.

Step 3: Analyze the Given Function

The given function is \(f(x) = 3^{x+1}\). Notice that this function contains \(x\) in the exponent. This is characteristic of an exponential function, not of a power or polynomial function.

Step 4: Determine the Correct Category

Since \(f(x) = 3^{x+1}\) does not have \(x\) as a base but rather as an exponent, it does not match the criteria for either a power function or a polynomial function. Therefore, it must be classified as neither of these.

Key Concepts

Function Classification

Classifying functions correctly is important in mathematics, especially when trying to understand or solve problems. Functions are essentially rules that assign each input exactly one output. They come in various forms, and recognizing these is the first step in function classification. When faced with a function, we determine whether it adheres to the definitions of well-known types such as polynomial, exponential, or power functions. Each category has distinct characteristics that can help us classify the function correctly. For example:

• A polynomial function has terms with variables raised to a non-negative integer power.
• A power function is simpler, with one term and a variable raised to a real number power.
• Exponential functions are the reverse of power functions, having the variable in the exponent.

Given the function \(f(x) = 3^{x+1}\), we see that \(x\) is in the exponent, making it an exponential function, and not fitting into the categories of power or polynomial functions.

Power Functions

Power functions are a specific type of mathematical function with a structure that's both straightforward and powerful. Defined by the formula \(f(x) = kx^n\), these functions consist of a constant multiplier \(k\) and an exponent \(n\) applied to the variable \(x\). This formula is easy to spot when \(x\) is used as the base of the exponent. Characteristics of power functions include:

• One term, where the variable base \(x\) is raised to any real number power \(n\).
• Non-complex expressions often result in straightforward calculations.
• If \(n\) is a negative number, it represents inverse relationships, like \(f(x) = x^{-1}\).

Power functions are often encountered when studying phenomena like gravity or inverse square laws. However, these functions differ from the exponential functions, where the variable is part of the exponent rather than the base.

Polynomial Functions

Polynomial functions are among the most common types of functions in algebra due to their versatile and adjustable nature. They are expressed as a sum of multiple terms, each consisting of a coefficient and a variable raised to a power. The general form of a polynomial function is \(f(x) = a_n x^n + a_{n-1} x^{n-1} + \,\cdots\, + a_0\), where each \(a_i\) is a coefficient, and the powers \(n\) are non-negative integers.Key features of polynomial functions include:

• Multiple terms with variable parts that maintain the variable base.
• The degree of the polynomial is the highest power of \(x\) that appears with a non-zero coefficient.
• They can be linear, quadratic, cubic, etc., depending on the highest exponent.

Unlike power functions with a single term, polynomials can be as simple as \(f(x) = x + 2\) or as complex as \(f(x) = 4x^5 + 3x^3 - 2x + 7\). Since the function \(f(x) = 3^{x+1}\) does not fit any of these forms, it cannot be classified as a polynomial function.