Question

Identify the inside function, \(u=g(x)\), and the outside function, \(y=f(u)\). $$ y=f(g(x)) \quad u=g(x) \quad y=f(u) $$ $$ y=\left(4-x^{2}\right)^{-1} $$

Short answer

For the function \(y=(4-x^{2})^{-1}\), the inside function is \(u=g(x)\) where \(g(x)=4-x^{2}\) and the outside function is \(y=f(u)\) where \(f(u)=u^{-1}\).

Step-by-step solution

Identify the Entire Function

The given function is \(y=(4-x^{2})^{-1}\). This is a composite function, meaning it is composed of two simpler functions, the inside and the outside function.

Identify the Inside Function

The inside function \(u=g(x)\) is found in the parentheses. That is, the function being acted upon first. In this case, it is \(u=x^{2}\). You can also notice it would fit into the mentioned form \(u=g(x)\). Here, \(g(x)=4-x^{2}\).

Identify the Outside Function

The outside function \(y=f(u)\) is the operation that is applied to the result of the inside function. In this case, it is \(f(u)=u^{-1}\). This function takes the output of the inside function (result of \(u\)) and applies the operation \(-1\) to it, essentially finding the reciprocal of the output of the inside function.

Key concepts

Function Composition

When we talk about function composition, we're looking at a process where one function is applied to the result of another function. Think of it as plugging one function into another. This concept is very handy in mathematics and many real-world applications.
For instance, if you have two functions, let's say, \( g(x) \) and \( f(u) \), composing them gives you a new function: \( f(g(x)) \). Essentially, we're taking the output of the first function \( g(x) \) and using it as the input for the second function \( f(u) \).
Composite functions are like building blocks. By combining simpler functions, you create a more complex function that can describe more intricate relationships or patterns.
Here are a few reasons why function composition is useful:

• It allows mathematicians to simplify complex problems into manageable parts.
• People in fields such as physics, economics, and computer science use it to model and analyze systems efficiently.

Remember, the key idea is to evaluate the inside function first before moving on to the outside function.

Inside Function

The inside function is often the first step in breaking down a composite function. It's the function you find within the parentheses and the function that gets calculated first. In our example, the inside function \( g(x) = 4 - x^2 \), captures this initial operation.
To find the inside function in a given composite function \( y=f(g(x)) \), look for the part where the primary action takes place. In the expression \( (4-x^2)^{-1} \), \( 4-x^2 \) is evaluated before any further operations.
It's useful to identify the inside function because:

• It helps in understanding the sequence of operations when dealing with complex equations.
• Knowing the inside function assists in tasks like differentiation using the chain rule in calculus.

By determining the inside function, you set the stage for applying the outside function appropriately, ensuring the operations reflect the intended mathematical process.

Outside Function

Once you've identified the inside function, it's time to figure out the outside function \( f(u) \). The outside function is applied after you evaluate the inside function. Using our example, once you compute \( u = 4 - x^2 \), the outside function is \( f(u) = u^{-1} \), which essentially takes the reciprocal.
This operation is crucial because the outside function determines how the result from the inside function is modified or transformed. It's similar to icing on a cake - the inside gives the cake it's form, but the outside brings the final touch.
The importance of recognizing the outside function includes:

• Facilitating the application of advanced calculus techniques, such as the chain rule.
• Helping in translating real-world scenarios and relationships into mathematical terms.

Understanding both functions in a composition allows for greater insight into how complex relationships are constructed and provides the tools necessary for further manipulation and analysis.