Question

In the form \(y=\) \(k \cdot(h(x))^{p}\) for some function \(h(x)\). $$ y=\frac{(2 x+1)^{5}}{3} $$

Short answer

In the given function $$y = \frac{(2x + 1)^5}{3}$$, identify the values of k, p, and the function $$h(x)$$ to rewrite the function in the form of $$y = k \cdot (h(x))^p$$.

Step-by-step solution

Identifying k

In the given function, $$y = \frac{(2x + 1)^5}{3}$$, the constant value multiplied to the power function is $$\frac{1}{3}$$. Thus, k is: $$k = \frac{1}{3}$$.

Identifying p

In the given function, we have the term $$(2x + 1)^5$$, which shows that the power to which the function is raised is 5. Thus, p is: $$p = 5$$.

Identifying h(x)

In the given function, the term inside the power function is $$(2x + 1)$$. Thus, the function h(x) is: $$h(x) = 2x + 1$$.

Rewrite the function in the form

Now we can rewrite the given function in the desired form $$y = k \cdot (h(x))^p$$ using the values we identified: $$y=\frac{1}{3}\cdot(2x+1)^5$$

Key concepts

Powers of Functions

When dealing with powers of functions, understanding how a function is raised to a certain power is crucial. Specifically, if you see an expression like \((h(x))^p\), it means that the function \(h(x)\) is multiplied by itself \(p\) times. In the example \((2x + 1)^5\), the function \(h(x) = 2x + 1\) is raised to the 5th power. This can be visualized as multiplying \((2x + 1)\) by itself 5 times.
This concept is essential when simplifying expressions, as it affects the overall behavior and values of the function, particularly over different domains of \(x\). In polynomial contexts, powers of functions explain how terms expand and shape the curve of the graph. Learning to recognize and manipulate powers of functions allows for a more in-depth understanding of advanced algebraic expressions and functions.

Function Notation

Function notation is a way to name a function and clearly express its input-output relationship. It's typically written as \(f(x)\) where \(f\) is the name of the function, and \(x\) is the variable. This notation is not limited to polynomials or linear expressions but applies across mathematical existence, offering a convenient way to represent function rules.
For the expression \(h(x) = 2x + 1\), \(h\) is the name of the function, and \(2x + 1\) is what you do to \(x\) to find the result of \(h(x)\).

• Benefits of function notation include clear communication of dependent and independent variables.
• It provides a universal language that simplifies understanding of complex operations.

In our context, clearly identifying \(h(x)=2x+1\) helped in rewriting and understanding the expression \(y=\frac{1}{3}\cdot(2x+1)^5\).

Constant Multiples

Understanding constant multiples is crucial when working with expressions like \(k \cdot(h(x))^p\). A constant multiple is simply a fixed number that is multiplied by a function or term throughout the expression. In the exercise solved, the constant \(k\) is \(\frac{1}{3}\), which means the entire function \((2x+1)^5\) is being divided or scaled by 3.
Having a constant multiple like \(k\) affected by multiplication or division influences the magnitude or steepness of the graph but does not change its fundamental shape.

• Constant multiples often aid in normalizing or scaling functions to fit desired results.
• They are integral in calculations involving derivatives and integrals as well.

Identifying these multiples helps streamline solving equations and offers insight into function behaviors.