Problem 7 - Applied Calculus

Question

Determine whether or not the function is a power function. If it is a power function, write it in the form \(y=k x^{p}\) and give the values of \(k\) and \(p\). $$y=3 \cdot 5^{x}$$

Step-by-Step Solution

Answer

The function is not a power function.

Step 1: Understand the Definition

A power function is a function of the form \( y = k x^p \), where \( k \) and \( p \) are constants, and the variable \( x \) is raised to a constant power \( p \).

Step 2: Analyze the Given Function

The given function is \( y = 3 \cdot 5^x \). Here, the variable \( x \) is in the exponent, which is characteristic of an exponential function, not a power function.

Step 3: Identify the Form

Compare \( y = 3 \cdot 5^x \) with \( y = k x^p \). In \( y = k x^p \), \( x \) must be the base raised to the power \( p \), but in \( y = 3 \cdot 5^x \), \( x \) is the exponent. Therefore, it does not match the form of a power function.

Key Concepts

Understanding Exponential Functions

An exponential function is a mathematical expression where the variable appears in the exponent. This is a key distinction that sets exponential functions apart from other types of functions, like power functions. The general form of an exponential function can be written as \( y = a \cdot b^x \), where:

• \( a \) is a constant coefficient and represents the initial amount or starting value.
• \( b \) is the base and it consists of a positive number other than 1, which dictates the growth or decay factor of the function.
• \( x \) is the exponent and is typically the independent variable.

Exponential functions are widely used in real-life scenarios, such as modeling population growth, radioactive decay, and interest computations in finance. A unique characteristic of exponential functions is their rapid increase or decrease, which makes them powerful in depicting scenarios that change swiftly.

Function Comparison

Function comparison involves examining and distinguishing different types of functions to understand their behavior and form. When assessing a function like \( y = 3 \cdot 5^x \), it's crucial to identify whether it is a power function or an exponential function. In a power function, the format \( y = k x^p \) means the variable \( x \) serves as the base and is raised to a constant power \( p \). Observing \( y = 3 \cdot 5^x \):

• \( x \) is the exponent—the defining trait of an exponential function—not the base.
• The base in this case, 5, is constant, further reinforcing that it does not fit the power function mold.

Function comparison helps in determining the proper categorization of a function, which is essential for solving related mathematical problems and applying them to real-world contexts. Recognizing the form determines how the function behaves mathematically.

The Concept of Constant Power

The term 'constant power' in mathematics refers to the fixed exponent in a power function, forming the expression \( y = k \cdot x^p \). In this configuration:

• \( k \) is a constant coefficient that influences the amplitude or scaling of the graph but not its basic shape.
• \( p \) is the constant power or exponent that determines the curvature or steepness of the graph.
• \( x \) acts as the base, meaning it is the main variable that is manipulated with the constant power.

In a power function, having a constant power implies that although the base \( x \) can vary, the exponent remains unchanged, which notably impacts the graph's nature and form. Constant power functions are prevalent in physics for expressing polynomial relationships and are instrumental in deriving solutions for equations where one quantity affects another non-linearly. By understanding these principles, one can surely differentiate a power function from other kinds of functions.