Question

In the form \(y=\) \(k \cdot(h(x))^{p}\) for some function \(h(x)\). $$ y=5 \sqrt{12-\sqrt[3]{x}} $$

Short answer

Question: Identify the values of \(k\), \(h(x)\), and \(p\) in the function \(y = 5\sqrt{12-\sqrt[3]{x}}\). Answer: For the given function, we have \(k = 5\), \(h(x) = 12-\sqrt[3]{x}\), and \(p = \frac{1}{2}\).

Step-by-step solution

Identify k, h(x), and p in the given function

In the given function, \(y = 5\sqrt{12-\sqrt[3]{x}}\), we can first identify \(k\) as the constant multiplier. In this case, \(k = 5\). Remember that \(\sqrt{x}\) can be written as \(x^{\frac{1}{2}}\). Thus, the given function can be rewritten as: $$ y = 5(12-\sqrt[3]{x})^{\frac{1}{2}}. $$ Now we can easily identify \(h(x)\) as the function inside the parentheses, \(h(x) = 12-\sqrt[3]{x}\). Lastly, we can now identify \(p\) as the power to which \(h(x)\) is raised. In this case, \(p = \frac{1}{2}.\) In conclusion, for the given function $$y = 5 \sqrt{12 - \sqrt[3]{x}},$$ we have \(k = 5\), \(h(x) = 12-\sqrt[3]{x}\), and \(p = \frac{1}{2}\).

Key concepts

Function Notation

Function notation provides a way to express relationships between variables in a systematic and clear manner. In essence, when you see an equation like \(y = f(x)\), it means that \(y\) is the output resulting from applying the function \(f\) to the input \(x\). This setup is crucial in understanding how variations in the input affect the output.

• **Function Definition**: Specifies how to convert an input into an output.
• **Functional Output**: The function produces this based on the given input.
• **Different Components**: In the function \(y = k \cdot (h(x))^p\), \(k\) is a constant, \(h(x)\) is another function, and \(p\) is the exponent.

Having a clear grasp of function notation allows for better interpretation and manipulation of these equations, aiding in various calculations and transformations.

Exponents and Radicals

Exponents and radicals are key concepts in algebra that express powers and roots respectively. An exponent such as \(x^n\) indicates that \(x\) is multiplied by itself \(n\) times. Conversely, a radical, like \(\sqrt{x}\), suggests finding a root of \(x\).

• **Understanding Exponents**: Represents repeated multiplication.
• **Connecting Radicals**: A radical is essentially an exponent in a fractional form. For instance, \(x^{\frac{1}{2}}\) is \(\sqrt{x}\).
• **Conversion**: Converting between these forms simplifies calculations. For example, converting \(\sqrt{12-\sqrt[3]{x}}\) to \((12-\sqrt[3]{x})^{\frac{1}{2}}\) helps identify components like \(p\).

Recognizing how to work with these expressions is vital for simplifying and solving algebraic equations.

Identifying Parameters in Functions

Locating parameters such as constants and exponents within a function is crucial for understanding its structure and behavior. In the equation \(y = k \cdot (h(x))^p\), identifying \(k\), \(h(x)\), and \(p\) reveals much about the function's form and potential transformations.

• **Constant \(k\)**: Often a multiplier affecting the function's overall growth or decay. For example, \(k = 5\) in \(y = 5 \sqrt{12-\sqrt[3]{x}}\) shows the strength of its impact.
• **Function \(h(x)\)**: The core expression within, like \(12-\sqrt[3]{x}\) here, dictates the primary behavior and internal transformation.
• **Exponent \(p\)**: The power to which \(h(x)\) is raised, influencing the degree of its transformation. Here, \(p = \frac{1}{2}\) implies a square root operation.

Understanding these parameters helps in analyzing and predicting the behavior of algebraic functions across different contexts.