Problem 13 - College Algebra

Question

For the following exercises, identify the function as a power function, a polynomial function, or neither. $$ 7-2 x^{2} $$

Step-by-Step Solution

Answer

The function is a polynomial function.

Step 1: Understand the given function

The given function is \( f(x) = 7 - 2x^2 \). We need to determine whether this is a power function, a polynomial function, or neither.

Step 2: Define Power Function

A power function has the form \( f(x) = kx^n \) where \( k \) is a constant and \( n \) is a real number. Each term is of the form \( kx^n \), with only one term present.

Step 3: Define Polynomial Function

A polynomial function has the form \( f(x) = a_nx^n + a_{n-1}x^{n-1} + \, ... + \, a_1x + a_0 \), where \( a_n, a_{n-1}, \,..., a_1, \ a_0 \) are constants, and \( n \) is a non-negative integer.

Step 4: Analyze the given function

The function \( f(x) = 7 - 2x^2 \) consists of two terms: \( 7 \) which is \( 7x^0 \) and \( -2x^2 \). Both terms fit the polynomial form \( a_nx^n \), where the powers of \( x \) are non-negative integers.

Step 5: Determine the type of function

Since the function \( f(x) = 7 - 2x^2 \) consists of terms fitting the criteria for a polynomial function, it is therefore a polynomial function. It does not fit the singular term requirement of a power function.

Key Concepts

Understanding Power Functions

Power functions are a fundamental concept in mathematics, particularly in algebra. They have a very specific form, which is expressed as \( f(x) = kx^n \). Here, \( k \) is a constant, which means it does not change, and \( n \) is a real number, which can be positive, negative, or zero. The defining feature of a power function is that it consists of exactly one term. This characteristic is crucial because, even if there are multiple powers of \( x \), it would no longer be a power function.

Consider the expression \( 3x^4 \). This is a power function because it has just one term in the form of \( kx^n \), with \( k = 3 \) and \( n = 4 \). If we added another term, such as \( + 7 \), it would no longer be a power function, but rather a polynomial. Understanding this distinction helps us categorize functions appropriately, as similar looking expressions might have different properties just based on structure.

Mathematics and Its Role in Power Functions

Mathematics provides the principles and frameworks that help us understand concepts like power functions. By studying mathematics, we gain the ability to see patterns, make connections, and understand the underlying rules that govern these expressions.

In the realm of power functions, mathematics allows us to explore how changes in the values of \( k \) and \( n \) affect the graph of the function. Here are some key insights:

• If \( n \) is positive, the graph increases as \( x \) increases when \( k > 0 \); if \( k < 0 \), the graph decreases.
• If \( n \) is negative, we deal with reciprocal functions, which have their own unique behaviors and properties.
• If \( n \) is zero, the function reduces to a constant function, \( f(x) = k \), which is simply a horizontal line on the graph.

These insights highlight how the mathematical properties of power functions inform their real-world behaviors and applications.

Delving into Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations that represent mathematical realities. Understanding them is vital to mastering more complex topics like polynomial functions.

A simple algebraic expression could be as basic as \( 2x + 3 \). This expression involves the variable \( x \), which can take on various values, and numbers, which are constants. In algebra, these expressions are used to represent general relationships and can be manipulated using various operations like addition, subtraction, multiplication, and division.

• When variables are raised to powers, the expressions become more complex, opening up advanced study areas like calculus and algebraic topology.
• Recognizing how different expressions fit into larger categories of mathematics helps in simplifying, combining, and solving equations, ultimately allowing for a deeper understanding of the subject.
• Being comfortable with algebraic expressions serves as a foundation for all future studies in mathematics, including working with functions like polynomials and power functions.

Understanding the structure and components of algebraic expressions is essential for tackling higher mathematics efficiently.