Problem 28 - Calculus with Concepts in Calculus

Question

Write \(h\) as the composite \(g \circ f\) of two functions \(f\) and \(g\) (neither of which is equal to \(h\) ). $$ h(x)=\left(1-x^{2}\right)^{3 / 2} $$

Step-by-Step Solution

Answer

Choose \(f(x) = 1-x^2\) and \(g(u) = u^{3/2}\).

Step 1: Identify Outer and Inner Functions

To express \( h(x) = (1-x^2)^{3/2} \) as a composite of two functions \( f \) and \( g \), determine a clear separation of the function into outer and inner parts. Notice that there's a squared term inside and an exponent outside. The inner function can be the expression inside the parentheses, while the outer function can be the exponentiation applied to that expression.

Step 2: Define the Inner Function \( f(x) \)

Let the inner function \( f(x) \) correspond to the expression inside the parentheses. Choose \( f(x) = 1 - x^2 \). This function simplifies the original expression by isolating the part \( 1 - x^2 \), which \( g \) will act upon.

Step 3: Define the Outer Function \( g(u) \)

Now define the outer function \( g(u) \) to transform the result of \( f(x) \) appropriately. Since we want to take \( (1 - x^2) \) raised to the power of \( \frac{3}{2} \), define \( g(u) = u^{3/2} \). This way, when applied to the outcome of \( f(x) \), we get the desired computation.

Step 4: Verify the Composition \( (g \circ f)(x) \)

Combine these functions to verify that they form \( h(x) \). Compute \( (g \circ f)(x) = g(f(x)) = g(1 - x^2) \). Substituting into \( g(u) \), we get \( (1 - x^2)^{3/2} \), which matches \( h(x) \). This confirms that \( h(x) \) is indeed the composite of \( g \) and \( f \).

Key Concepts

Outer Function

The outer function is like the cover of a book. In our exercise, the outer function is the author applying the final touch to the storyline. It operates on the result provided by the inner function. We define it to handle the final computation, giving the expression its ultimate shape.
In simpler terms, the outer function takes the output of the inner function and performs further operations on it.
In our exercise:

• The outer function is defined as \( g(u) = u^{3/2} \).
• It takes whatever output the inner function produces and raises it to the power of \( \frac{3}{2} \).

By applying \( g(u) \), we're transforming the input in a way that fits our needed result for \( h(x) \). The outer function is key to shaping the behavior of the composite function, directing what happens to the inner function’s output.

Inner Function

The inner function nests within the composite function like a story's core within a book. It's the initial component that the outer function manipulates.
When we talk about looking inside, the inner function provides the groundwork that triggers the entire operation.
In this exercise, let's consider:

• The inner function is defined as \( f(x) = 1 - x^2 \).
• This function crafts the primary input for the outer function.

Imagine it as the foundation. \( f(x) \) sets up the expression inside the parentheses. Without defining the inner function correctly, the overall computation would turn out flawed, much like a story without a solid plot. Properly understanding \( f(x) \) means recognizing its role as a crucial step before any further transformation by the outer function.

Composite Function

The composite function is the blend of both outer and inner functions, seamlessly intertwined, much like the ingredients of a cake coming together into a delicious creation. It embodies the whole process from start to finish. This is where we see the combination of \( g \) and \( f \) work harmoniously to achieve the desired \( h(x) \).
The composite function is described by the notation \( (g \circ f)(x) \), which is read as "\( g \) composed with \( f \)." This expression denotes that you're applying \( f(x) \) first to get an intermediate value, and then applying \( g \) to that result.
In this problem, the composite function is:

• \((g \circ f)(x) = g(f(x)) = g(1 - x^2) = (1 - x^2)^{3/2}\).
• It shows the collaboration between \( g \) and \( f \) to produce \( h(x) \).

Think of this unity as a team working together in perfect synchronization. Each function contributes its part, ensuring that the final result mirrors \( h(x) = (1-x^2)^{3/2} \). It's an elegant dance between two processes, creating a new function through their combination.