Problem 36 - 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.) $$ f(x)=x^{4} $$

Step-by-Step Solution

Answer

The function \(f(x) = x^4\) is a polynomial function.

Step 1: Define a Polynomial Function

A polynomial function is a type of mathematical expression that consists of variables and coefficients. The variables have non-negative integer exponents, and the function can be written in the form:\[ f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0,\]where \(a_n, a_{n-1}, \ldots, a_0\) are coefficients and \(n\) is a non-negative integer.

Step 2: Analyze the Given Function

The function provided is \(f(x)=x^{4}\). In this function, \(x\) is raised to a power, and it follows the form of a polynomial because its only term is \(x^4\), which has a non-negative integer exponent, specifically \(n = 4\).

Step 3: Classify the Function

Since \(f(x)=x^{4}\) fits the criteria of a polynomial function, we classify it as such. It doesn't contain variables in the denominator, fixed growth rates, or piecewise conditions, so it's not a rational, exponential, or piecewise linear function.

Key Concepts

Mathematical Expressions

Mathematical expressions are combinations of numbers, symbols, and operators used to represent values or relationships. In the realm of polynomial functions, expressions often include variables raised to different powers, known as exponents. A polynomial expression specifically includes terms where the variables are raised to non-negative integer exponents. This makes it easy to identify the behavior and properties of the function. Understanding mathematical expressions is key to grasping the structure and classification of functions like polynomials.

• Variables: symbols that can take various values, usually represented by letters such as \(x\) or \(y\).
• Operators: symbols that represent mathematical operations like addition \(+\), subtraction \(-\), multiplication \(\times\), and division \(\div\).

By recognizing these parts of expressions, you'll better understand how they form more complex mathematical relationships.

Integer Exponents

Exponents are a way to express repeated multiplication of a number. In a polynomial function, the exponents applied to variables are always non-negative integers. This means the exponents are whole numbers like 0, 1, 2, 3, and so on. For example, in the polynomial expression \(f(x)=x^{4}\), the exponent is 4. This tells us that the variable \(x\) is multiplied by itself 4 times: \(x \times x \times x \times x\).

• Non-negative: Exponents are never negative in polynomials, ensuring the function remains smooth and well-defined.
• Whole numbers: Only whole numbers (0, 1, 2...) are valid exponents in polynomials.

Understanding integer exponents helps in identifying polynomial functions and their degree, which is the highest power of the variable in the expression.

Coefficients

Coefficients are the numerical factors that multiply the variables in a polynomial expression. They play a crucial role in determining the overall size and direction of a polynomial's terms. In a general polynomial form \(f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0\), the \(a_n, a_{n-1}, \ldots, a_0\) are the coefficients. For example, in the expression \(f(x)=x^{4}\), though seemingly absent of a number in front of \(x^4\), the coefficient is actually 1, as it can be rewritten as \(1\times x^4\).

• Impact: Coefficients determine how steep or flat the graph of the function appears.
• Non-zero: At least one coefficient must be non-zero for it to qualify as a polynomial term.

Appreciating the role of coefficients can help clarify how different polynomial terms contribute to the function's graph and overall behavior.