Problem 40 - 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.) $$ f(x)=x^{2}+9 $$

Step-by-Step Solution

Answer

The function \( f(x) = x^2 + 9 \) is a polynomial function.

Step 1: Understanding the Terms

To identify the type of function, we need to understand the characteristics of each type of function: - A polynomial function is of the form \( P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0 \), where all powers of \( x \) are non-negative integers.- A rational function is the ratio of two polynomials.- An exponential function has the variable in the exponent, like \( a^x \).- A piecewise linear function is defined by different linear expressions over different intervals of the domain.

Step 2: Analyzing the Given Function

The function given is \( f(x) = x^2 + 9 \). It consists of a single term with \( x^2 \) and a constant term 9, making it a sum of terms with non-negative integer powers of \( x \).

Step 3: Checking Polynomial Criteria

The power of \( x \) in \( f(x) = x^2 + 9 \) is 2, which is a non-negative integer. Additionally, the function does not have any variables in the denominator or as exponents, it is not a ratio of two polynomials, nor does it have piecewise definitions.

Step 4: Conclusion of the Function Type

Since \( f(x) = x^2 + 9 \) meets the criteria of a polynomial function and none of the others (rational, exponential, or piecewise linear), we can conclude it is a polynomial function.

Key Concepts

Function Identification

Identifying a function involves examining the structure and components present in the mathematical expression. Each type of function has unique characteristics that distinguish it from others. For instance, a polynomial function consists solely of terms where variables are raised to whole number powers, while a rational function is a fraction composed of two polynomial expressions. By inspecting these features, you can determine the function's type without needing to graph it.

To identify the function given in the exercise, which is \( f(x) = x^2 + 9 \), we need to observe the components of the expression. We look at the power of \( x \) and the arrangement of terms, like whether the expression includes division or exponential growth components. This systematic identification process helps wrap our minds around the function's nature and how it behaves within mathematics' broader landscape.

Polynomial Criteria

A function is considered a polynomial if it matches specific criteria related to the powers of its variables. In a polynomial function, all powers of the variable \( x \) are non-negative integers. This means there are no variables in denominators, nor are there any negative or fractional exponents.

To see if \( f(x) = x^2 + 9 \) fits the polynomial criteria, we check the following:

• Each term must have non-negative integer exponents: In our example, \( x^2 \) is valid.
• No division by variables: The given expression doesn't divide by \( x \).
• There should be no variables in the exponents: Here, \( x^2 \) satisfies this.
• Absence of piecewise terms: The expression is uniform across all \( x \).

This function clearly meets all the polynomial criteria, thus confirming it as a polynomial function and not any other type such as rational or exponential. It doesn't involve fractional terms, exponential growth, or a piecewise structure.

Types of Functions

Functions can be classified into several types based on their structural characteristics. Understanding these differences is essential for correctly interpreting a function's behavior.• Polynomial functions, like our \( f(x) = x^2 + 9 \), are defined by sums of powers of a variable without division or negative powers.

• Rational functions involve ratios of polynomial expressions \( \left( \frac{P(x)}{Q(x)} \right) \).
• Exponential functions have variables in the exponent, evident in forms like \( a^x \).
• Piecewise functions comprise multiple sub-function expressions, each applicable over different intervals.

Each distinct type of function has specific rules guiding how it behaves and what kind of mathematical problems it can help solve. By recognizing these characteristics, we can accurately categorize and leverage functions to model real-world phenomena or solve theoretical challenges in mathematics.