Problem 4 - 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 \(p\) $$y=\frac{3}{x^{2}}$$

Step-by-Step Solution

Answer

Yes, it's a power function with \(k = 3\) and \(p = -2\).

Step 1: Identifying the form of the function

The given function is \(y = \frac{3}{x^2}\). To determine if this is a power function, we need to see if we can express it in the form \(y = kx^p\).

Step 2: Rewriting the function

The current expression can be rewritten by moving the \(x^2\) from the denominator to the numerator as: \(y = 3x^{-2}\). This expression is now in the form \(y = kx^p\) where \(k = 3\) and \(p = -2\).

Step 3: Conclusion

Since we were able to rewrite the expression in the form \(y = kx^p\), the function \(y = \frac{3}{x^2}\) is indeed a power function.

Key Concepts

Function Identification

Identifying if a function is a power function involves recognizing whether it can be rewritten in the specific form of a power function, which is generally expressed as \( y = kx^p \). In this format:

• \( y \) is the dependent variable.
• \( x \) is the independent variable.
• \( k \) is a constant coefficient.
• \( p \) is the power or exponent of \( x \).

To determine if a given function, such as \( y = \frac{3}{x^2} \), can be categorized as a power function, we must examine if it can be transformed into this standard format. This involves utilizing algebraic manipulations, such as moving terms across the fraction line or altering powers, to achieve the necessary structure. Successfully rewriting a function like \( y = \frac{3}{x^2} \) into \( y = 3x^{-2} \) confirms it follows the form of a power function with specific values for \( k \) and \( p \).

Rewriting Functions

Rewriting a function often involves changing its original form to another representation without altering its essential meaning or properties. This is particularly useful in the context of power functions, where you may start with a function in a different algebraic form and need to express it as \( y = kx^p \).
For example, in the function \( y = \frac{3}{x^2} \), the goal is to express it as a power function. By moving \( x^2 \) from the denominator to the numerator, you change the form to \( y = 3x^{-2} \):

• The exponent changes sign when moving from the denominator to the numerator, transforming \( x^{2} \) into \( x^{-2} \).
• This new expression clearly fits the form \( y = kx^p \), with \( k = 3 \) and \( p = -2 \).

Rewriting functions is a common process in algebra, which helps to simplify expressions and easily identify their characteristics. By maintaining equivalent expressions, like \( y = \frac{3}{x^2} \) and \( y = 3x^{-2} \), we ensure the accuracy of the solution while providing clarity and uniformity in the expression.

Algebraic Expressions

An algebraic expression is a mathematical phrase involving numbers, variables, and operation symbols that model a particular relationship or rule. These expressions can be created, interpreted, and rewritten depending on the need to solve a problem or identify specific characteristics of a function.
In the case of power functions, recognizing and manipulating algebraic expressions allows one to express functions in a more useful form that highlights their key attributes. Consider the function \( y = \frac{3}{x^2} \):

• This is a rational expression where \( 3 \) (numerator) is divided by \( x^2 \) (denominator).
• Algebraic operations can alter this into a power function: \( y = 3x^{-2} \).
• By rewriting \( x^2 \) as \( x^{-2} \), we maintain the essential mathematical properties while aligning the expression with standard power function form.

Algebraic expressions are foundational in mathematics, enabling the rewriting process that transforms complex or unfamiliar expressions into standard forms. This makes it easier to analyze and apply mathematical concepts, such as identifying values of coefficients and exponents in power functions.