Problem 53 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 53-60: Find a composite function form for \(y\). \(53 y=\left(x^{2}+3 x\right)^{1 / 3}\) $$ y=\left(x^{2}+3 x\right)^{1 / 3} $$

Step-by-Step Solution

Answer

The composite function is \( y = f(g(x)) \) with \( f(u) = u^{1/3} \) and \( g(x) = x^2 + 3x \).

Step 1: Identify Inner Function

The composite function can be expressed as \( y = f(g(x)) \). First, identify the inner function \( g(x) \). Here, the expression inside the cube root is \( x^2 + 3x \). Hence, let \( g(x) = x^2 + 3x \).

Step 2: Determine Outer Function

Once the inner function has been identified, determine the outer function. Knowing \( y = (x^2 + 3x)^{1/3} \), the outer function is a cube root function. Thus, \( f(u) = u^{1/3} \), where \( u = g(x) \).

Step 3: Express Composite Function

Using the identified functions \( f(u) = u^{1/3} \) and \( g(x) = x^2 + 3x \), the composite function can be written as \( y = f(g(x)) = (g(x))^{1/3} \). This simplifies to \( y = (x^2 + 3x)^{1/3} \).

Key Concepts

Inner Function

In the realm of composite functions, the inner function is one that sits neatly inside another function, essentially serving as the initial step in our operation chain. To find the inner function, we look for the expression that is surrounded by operation or transformation signs. In the exercise given, the expression \(x^2 + 3x\) is what we're focusing on, as it is enclosed within the cube root.

The inner function, which we'll denote as \(g(x)\), is formulated as follows:

• It's like the foundation of a house, as it forms the primary structure before any further calculation occurs.
• We have chosen \(g(x) = x^2 + 3x\) because this expression is the core of our composite function, where all inputs initially interact.

Understanding the inner function sets the stage for determining which operation to perform next, achieving a stepwise, logical calculation process.

Outer Function

The outer function acts like a wrapper or a layer around the inner function, guiding us on what to do with the outcome of the initial calculation. Once the inner function is processed, the outer function dictates the next step in our formula or expression. In our case, the expression that follows the initial \(g(x)\) is the cube root, represented as raised to the power of one-third.

This means our outer function, denoted as \(f(u)\), can be expressed as:

• \(f(u) = u^{1/3}\), where \(u\) is the result of \(g(x)\).
• Think of it as a filter that modifies the output given by our inner function, providing an all-encompassing perspective of how transformations occur in a sequence.

Recognizing the outer function's role is crucial for visualizing the whole pathway from input to the final result in a composite function form.

Cube Root

The cube root is a special mathematical operation used to find a number that, when multiplied by itself twice (cubed), results in the original number. It acts inversely to the process of cubing a number. When you see an expression like \(u^{1/3}\), this indicates that you are looking to find the cube root of \(u\).

In our composite function problem, the cube root plays a pivotal role as it forms part of our outer function:

• It modifies the output of \(g(x) = x^2 + 3x\), transforming it into the final \(y\) value.
• The cube root is integral to unraveling relationships within data sets, providing a simplified version of potentially complex multiplication layers.

In composite functions, applying the cube root serves as a bridge from operation to solution, displaying how transformations can unite different mathematical expressions into one concise result.

Function Composition

Function composition refers to the process of combining two functions such that the output from the first function becomes the input for the second function. In our instance, this is a process where \(y = f(g(x))\), where we first operate with \(g(x)\) and then apply \(f(u)\) to the result.

This seamless integration of functions allows for constructing complex expressions from simpler ones. Let's break this into our provided example:

• The inner function is \(g(x) = x^2 + 3x\).
• The outer function modifies it: \(f(u) = u^{1/3}\).
• Thus, the overall composition or combination results in \(y = (x^2 + 3x)^{1/3}\).

Understanding function composition is like learning to layer inputs efficiently while processing them through sequential transformations, resulting in rich, multifaceted mathematical operation chains.