Problem 2 - 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 is a power function with \(k = 3\) and \(p = -2\).

Step 1: Identify Form of Function

The given function is \(y = \frac{3}{x^2}\). This function can be rewritten to fit the form of a power function by expressing \(x^2\) in the numerator.

Step 2: Rewrite Function

To convert \(y = \frac{3}{x^2}\) into a power function, we can rewrite it as \(y = 3x^{-2}\). This step involves understanding that \(x^{-2} = \frac{1}{x^2}\).

Step 3: Identify Power Function Form

A power function can be expressed in the form \(y = kx^p\). Comparing \(y = 3x^{-2}\) with this form, we see that \(k = 3\) and \(p = -2\).

Key Concepts

Function Identification

Identifying whether a function is a power function is a critical first step. A power function is a specific type of algebraic function of the form \( y = kx^p \). This means it can be expressed with a constant \( k \) and an exponent \( p \). To determine if a given function fits this form, inspect the relationship between the dependent variable \( y \) and the independent variable \( x \). In our example, we have the function \( y = \frac{3}{x^2} \). At first glance, this does not immediately appear to be in the form of \( kx^p \). Hence, further investigation is needed to determine if it's expressible as a power function.

Function Rewriting

Rewriting functions is a useful technique to unveil hidden structures like power functions. The original function \( y = \frac{3}{x^2} \) can be challenging to classify directly because it's presented as a fraction. However, using algebraic manipulation, we can rewrite this expression to fit the power function format. Recall the property \( \frac{1}{x^2} = x^{-2} \). Applying this understanding, we rewrite the function as \( y = 3x^{-2} \), clearly exhibiting a power function structure. Rewriting functions like this helps in revealing the parameters \( k \) and \( p \) that classify it precisely as a power function.

Exponent Rules

Understanding exponent rules is essential when dealing with power functions. Exponents allow us to represent repeated multiplication compactly. The rule \( a^{-n} = \frac{1}{a^n} \) was key in converting the original function into a power function. By applying this rule, the denominator \( x^2 \) in the expression \( \frac{3}{x^2} \) is transformed to \( x^{-2} \), changing the overall function to \( y = 3x^{-2} \). Remember that exponent rules can simplify complex looking expressions, making it easier to identify function types and perform algebraic manipulations.

Parameter Identification

Once a function is expressed in the power function form, identifying the parameters \( k \) and \( p \) becomes straightforward. These parameters hold particular significance: \( k \) represents the constant coefficient scaling the function, while \( p \) indicates the power to which \( x \) is raised. In the function \( y = 3x^{-2} \), \( k = 3 \) and \( p = -2 \). Recognizing these parameters allows us to understand the behavior and properties of the power function, such as growth, decay, and its graph's general shape. Identifying \( k \) and \( p \) is the concluding step in confirming the function type and analyzing its characteristics.