Question

In the form \(y=\) \(k \cdot(h(x))^{p}\) for some function \(h(x)\). $$ y=\frac{2}{\sqrt{1+\frac{1}{x}}} $$

Short answer

Question: Convert the given function $$y=\frac{2}{\sqrt{1+\frac{1}{x}}}$$ into the form \(y=k \cdot(h(x))^{p}\) and identify the values of \(k\), \(h(x)\), and \(p\). Answer: The given function can be converted into the form \(y=k \cdot(h(x))^{p}\) as follows: $$y = 2 \cdot \left(\frac{x}{x+1}\right)^1$$. The values are \(k=2\), \(h(x)=\frac{x}{x+1}\), and \(p=1\).

Step-by-step solution

Rewrite the Function

Since the goal is to rewrite the given function into the form \(y=k \cdot(h(x))^{p}\), let's rewrite the function to isolate \(x\) in the denominator: $$y=\frac{2}{\sqrt{1+\frac{1}{x}}}=\frac{2}{\sqrt{\frac{x+1}{x}}}$$

Remove the Square Root

To eliminate the square root, raise both the numerator and the denominator to the power of 2: $$y= 2 \cdot(\frac{x}{x+1})$$

Identify the Values of k, h(x), and p

In the rewritten function \(y= 2 \cdot(\frac{x}{x+1})\), we can see that: - \(k=2\) - \(h(x)=\frac{x}{x+1}\) - \(p=1\) So, the given function can be written in the form \(y=k \cdot(h(x))^{p}\) as: $$y = 2 \cdot \left(\frac{x}{x+1}\right)^1$$

Key concepts

Function Transformation

Function transformation involves changing the position, shape, or size of a function's graph in a coordinate plane. Transformations can be applied through various techniques such as translations, reflections, dilations, and rotations of the function. Think of it as moving or changing the graph without altering its core characteristics. This concept helps us understand how algebraic expressions can alter and manipulate fundamental shapes of graphs within the coordinate system.

When transforming a function, typically you adjust its equation to stretch, compress, shift, or reflect its graph. For example, multiplying a function by a constant can stretch or compress it vertically or horizontally. Adding or subtracting from the input or output of a function shifts it up, down, left, or right.

When considering the exercise, we deal with expressing a complex function in the form of a simpler transformation using the format \( y = k \cdot (h(x))^{p} \). Each part of this form—\( k \), \( h(x) \), and \( p \)—plays a distinct role in transforming the original function. Here, \( k \) acts as a vertical stretch or compression factor, \( h(x) \) represents the inner function determining the main variable dynamics, and \( p \) dictates the power that transforms the function logarithmically or exponentially.

Rational Functions

Rational functions are fractions of polynomials, essentially the ratio of two polynomial functions. They often appear in mathematics as \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials. Understanding rational functions is crucial since they model many real-world processes like rates, concentrations, and economic behavior.

Key characteristics of rational functions include:

• Asymptotes: Lines that the graph approaches but never touches. Vertical asymptotes occur where the denominator is zero, while horizontal asymptotes are determined by the degrees of the polynomials involved.
• Discontinuities or "holes" where the function is not defined.
• End behavior that can either plateau or continue indefinitely based on the degree differences between \( P(x) \) and \( Q(x) \).

In the context of our exercise, the original function \( y=\frac{2}{\sqrt{1+\frac{1}{x}}} \) represents a rational function because it embodies polynomial operations, influenced by an additional radical component. Rewriting it in a simpler form \( y = 2 \cdot \left(\frac{x}{x+1}\right)^1 \) involves reducing it to a more standard rational function, showcasing function dynamics with polynomial inputs and outputs.

Exponentiation

Exponentiation is an essential mathematical operation that involves raising a number, known as the base, to a power, which is the exponent. This concept is critical in various fields, from solving algebraic equations to modeling exponential growth in real-world phenomena such as population dynamics and financial growth.

To understand exponentiation, consider \( a^b \) where \( a \) is the base and \( b \) is the exponent. This expression means multiplying \( a \) by itself \( b \) times. For example, \( 3^2 = 3 \times 3 = 9 \). Exponentiation can also involve powers that are fractions or even negatives, such as square roots and reciprocals.

In our problem, exponentiation aids in simplifying the expression and converting it into the desired form \( y = k \cdot (h(x))^{p} \). By recognizing the roles of the components \( k \), \( h(x) \), and \( p \), exponentiation allows us to elegantly express \( y \) in a calculated way. Here, \( p \) determines the degree of the transformation either as a direct representation (linear) or through more complex manipulations such as squaring, indicating the dynamics of the function's behavior.