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

Step-by-Step Solution

Answer

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

Step 1: Recognize the Form

The function given is \( f(x) = x^4 \). It appears in the form of \( x^n \), where \( n \) is a non-negative integer.

Step 2: Define Polynomial Function

A polynomial function is expressed as \( f(x) = a_nx^n + a_{n-1}x^{n-1} + \, ... \, + a_1x + a_0 \), where \( a_n, a_{n-1}, \, ... \, , a_0 \) are constants and \( n \) is a non-negative integer.

Step 3: Compare with Polynomial Definition

The given function \( f(x) = x^4 \) fits the definition of a polynomial function with \( a_4 = 1 \) and \( a_3, a_2, a_1, a_0 = 0 \). The term \( x^4 \) confirms it is a polynomial of degree 4.

Step 4: Confirm Function Type

Since \( f(x) = x^4 \) fits the form of a polynomial function as described, it is indeed a polynomial function.

Key Concepts

Degree of Polynomial

The degree of a polynomial is a key concept in understanding and categorizing polynomial functions. The degree is the highest power of the variable present in the polynomial. For example, in the polynomial \( f(x) = x^4 \), the term with the highest power is \( x^4 \). Thus, the degree of this polynomial is 4.

Why is this important? Because the degree tells us a lot about the polynomial! It helps in predicting the polynomial's behavior and sketching its graph. Moreover, it indicates:

• The maximum number of roots the polynomial can have.
• The number of turns in its graph.
• The end behavior of the graph; for instance, whether it rises or falls as the variable moves towards infinity.

Remember, in any polynomial function, the degree is always a non-negative integer, and it gives the polynomial its name: a degree of 4 makes it a "quartic" polynomial.

Non-Negative Integer Exponent

When we talk about polynomials, the term "non-negative integer exponent" frequently appears. Simply put, an exponent is the power to which the variable is raised in each term of the polynomial. In polynomial functions, these exponents are always non-negative integers.

This means they are whole numbers and include zero or any whole number greater than zero. Let's see some examples:

• The term \( x^4 \) has an exponent of 4, which is a non-negative integer.
• \( x^0 \) simply equals 1 (because anything raised to the power of zero is 1), and thus also fits the polynomial description.

Why can't we have negative or fractional exponents in polynomials? Because that would change the function type to something else, such as a rational function or a different form of expression altogether. In keeping with integer exponents, polynomials remain simple and easier to manage.

Polynomial Expression

A polynomial expression is a mathematical phrase that can involve a sum of terms, each consisting of a variable raised to a non-negative integer exponent and multiplied by a constant coefficient.

The expression \( f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \) defines a polynomial, where each \( a_i \) is a coefficient and \( x^i \) is the term with a non-negative integer exponent.

• The coefficients such as \( a_n, a_{n-1}, \ldots, a_0 \) dictate the size and direction of the polynomial's graph at different points.
• Constant terms like \( a_0 \) act as the starting point when the variable equals zero.

The beauty of polynomial expressions lies in their simplicity and the complexity they can portray. They can model a wide range of real-world situations and provide a foundation for many mathematical concepts. Understanding polynomial expressions helps in a deep exploration of algebraic operations and functions.