Problem 5 - 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 \(p\) $$y=2^{x}$$

Step-by-Step Solution

Answer

The function is not a power function.

Step 1: Identify the form of a power function

A power function is typically expressed in the form \( y = k x^p \), where \( k \) and \( p \) are constants, and \( x \) is the variable.

Step 2: Analyze the given function

The given function is \( y = 2^x \). Here, the variable \( x \) is in the exponent, which does not match the standard form of a power function where the variable \( x \) should be the base.

Step 3: Determine if the function is a power function

Since the variable \( x \) is an exponent rather than a base, \( y=2^x \) is not a power function according to the definition provided in Step 1.

Key Concepts

Exponential Function

When we talk about an exponential function, we are referring to a specific type of function where the variable, often denoted as \( x \), appears as an exponent. This is different from many other mathematical functions where the variable is typically the base. A classic example of an exponential function is \( y = a^{x} \), where \( a \) is a constant.

• One key feature of exponential functions is their rapid rate of growth or decay, depending on the base \( a \).
• If \( a > 1 \), the function grows as \( x \) increases.
• If \( 0 < a < 1 \), the function decreases as \( x \) increases.

For example, in the function \( y = 2^x \), the variable \( x \) is the exponent with the base being \( 2 \), making it an exponential function. This has many practical applications, from calculating compound interest to modeling population growth.

Function Analysis

Function analysis is a method used to study the behavior of functions, including their type, growth, or decline. By understanding the nature of a function, you can predict how changes in the variable will impact the overall output of the function.

• One aspect of function analysis is determining whether a given function is a specific type, such as a power function or an exponential function, as seen in the exercise.
• This involves comparing the given function's form with known standard forms.

In the provided solution, we performed function analysis by comparing the given function \( y = 2^x \) to the form of a power function \( y = k x^p \). By identifying that \( x \) was an exponent rather than a base, we concluded that it did not fit the power function type.

Mathematical Expressions

Understanding mathematical expressions is essential for solving problems and analyzing functions. A mathematical expression is a combination of numbers, variables, and operations that represent a specific value or values when evaluated.

• Expressions can come in different forms, including polynomials, rational expressions, and exponents, as seen with exponential functions.
• They serve as building blocks for more complex equations and functions.

In the analysis of functions, recognizing the role of each part of a mathematical expression helps to determine the type of function it represents. For instance, in \( y = 2^x \), recognizing that \( x \) is an exponent helps identify this as an exponential expression rather than a power function. By breaking down expressions into their components, you can better understand and solve more complicated mathematical problems.