Problem 33 - Applied Calculus

Question

33-46. 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.) $$ f(x)=x^{5} $$

Step-by-Step Solution

Answer

The function \( f(x) = x^5 \) is a polynomial function of degree 5.

Step 1: Identify the Function Type

To determine the type of function, observe the expression given: \( f(x) = x^5 \). This expression involves the variable \( x \) raised to the power of 5.

Step 2: Recognize Polynomial Characteristics

A polynomial function is characterized by the presence of terms that are sums or differences of variables raised to whole number powers, each multiplied by coefficients from the set of real numbers. In this case, \( x^5 \) is a term that fits this description.

Step 3: Determine the Function Degree

In a polynomial function, the degree is the highest power of the variable. Here, the highest power is 5. Therefore, \( f(x) = x^5 \) is a polynomial of degree 5.

Step 4: Conclude the Function Type

Since the function consists solely of \( x \) raised to a whole number power, \( f(x) = x^5 \) is identified as a polynomial function, specifically of degree 5.

Key Concepts

Function Identification

Identifying function types can help you understand their behavior and applications. To determine whether a given function is a polynomial, it's critical to recognize specific characteristics.
A polynomial function comprises terms that are sums or differences of variables raised to whole number powers. Each term has a coefficient, which is typically a real number.
For instance, the function given is \( f(x) = x^5 \), and it is a polynomial because it includes just one term where the variable \( x \) is raised to the power of 5. This specific form fits perfectly within the characteristics of polynomial functions because:

• All powers are whole numbers (not fractions or negatives).
• The coefficients are real numbers.

Recognizing a function's structure is key. So remembering these characteristics makes identifying polynomial functions straightforward.

Polynomial Degree

The degree of a polynomial is a fundamental feature. The degree tells us the highest power of the variable in the polynomial.
For example, in the function \( f(x) = x^5 \), the degree is 5 because the variable \( x \) is raised to the fifth power.
Understanding the degree of a polynomial function provides insight into its behavior.

• A higher degree indicates that the function's graph will have more curves.
• The degree directly impacts the function's maximum number of roots or intersections with the \( x \)-axis.
• It also reveals the number of turning points or peaks and troughs the graph might have.

Recognizing the degree of a polynomial instantly gives clues about the function's complexity and graph geometry, making it essential for further analysis.

Mathematical Analysis

Analyzing mathematical expressions like polynomial functions helps us predict their behavior and solve complex problems. Mathematical analysis involves examining expressions to understand their nature and constraints.
When looking at a polynomial like \( f(x) = x^5 \), mathematical analysis can be simplified to:

• Identify the impact of individual terms on the function's shape.
• Analyze the effect of the highest degree on the function's leading term, translating to how it behaves as \( x \) approaches infinity.
• Consider symmetry and calculus-based tools to evaluate turning points and inflection points.

The simplification and dissection of polynomial functions through analysis equip students to handle more complicated expressions logically and methodically, preparing them for tackling broader mathematical challenges.