Question

Identify the inside function, \(u=g(x)\), and the outside function, \(y=f(u)\). $$ y=f(g(x)) \quad u=g(x) \quad y=f(u) $$ $$ y=\left(x^{2}-2 x+3\right)^{3} $$

Short answer

The inside function, \(u=g(x)\), is \(x^{2}-2 x+3\), and the outside function, \(y=f(u)\), is \(u^{3}\).

Step-by-step solution

Identify the Inside Function

The inside function is the function within the brackets. In this case, the expression \(x^{2}-2 x+3\) is in the bracket, which makes it the inside function, denoted as \(u=g(x)\). So, \(u=x^{2}-2 x+3\).

Identify the Outside Function

The outside function is the function acting on the inside function. Here, the outside function is the cubing function. So if we substitute \(u=g(x)\) into our function, our function \(y=f(u)\) becomes \(y=u^{3}\). This shows that \(f(u)=u^{3}\).

Combine Both Functions

Now, we combine both the inside function and the outside function back into the original function, resulting in \(y=(x^{2}-2 x+3)^{3}\).

Key concepts

Inside Function

When we deal with composite functions such as y=f(g(x)), the inside function is the first operation that is applied to the input value, x. It's like the initial part of a manufacturing process where we begin shaping the raw material. In the exercise y=(x^{2}-2 x+3)^{3}, the inside function is u=g(x)=x^{2}-2 x+3. This is the heart of the compound function that gets 'packaged' by the outside function.

Understanding the inside function is crucial because the success of further operations depends on it. Just as a flaw in the first step of a production line can lead to a faulty product, any mistake in identifying or using the inside function will result in an incorrect overall function. Thus, identifying the inside function correctly allows for proper manipulation and simplification of complex expressions.

Outside Function

In contrast, the outside function takes the output of the inside function and applies another layer of operation to it — it's the finishing touch. Referring to our earlier example, once we have u=g(x), we then apply the outside function f(u)=u^{3}, which cubes the result of the inside function.

The outside function reshapes the preliminary 'product', which came from the inside function, into a final form. If we envision this as an assembly line, the outside function represents the stages where features and specifics are added, refining the product to meet its final specifications. So it's equally important to understand the role of the outside function in order to unravel or construct a composition of functions.

Function Operations

When we think about function operations, it refers to the various procedures we can perform with functions, including addition, subtraction, multiplication, division, and composition. Composition of functions, as shown in the exercise with y=(x^{2}-2 x+3)^{3}, is a special kind of operation that combines two functions in a way where the output of the first function (inside function) becomes the input of the second function (outside function).

It is important to handle these operations correctly, as the domain of the composite function is determined by both the domain of the inside function and the condition that the output of the inside function must lie in the domain of the outside function. Picture each operation as a tool that modifies the function in a specific way, fitting different pieces together to create an intricate machinery capable of producing diverse and complex outcomes.

Chain Rule

The chain rule is a principle in calculus that allows us to find the derivative of composite functions. When we have two linked processes like y=(x^{2}-2 x+3)^{3}, the chain rule gives us a way to measure the rate of change (derivative) of the entire operation relative to its individual parts.

This involves taking the derivative of the outside function and multiplying it by the derivative of the inside function. The rule can be summarized by the phrase 'the derivative of the outside with respect to the inside, times the derivative of the inside with respect to x.' In essence, the chain rule helps in navigating through the layers of operations by giving us a systematic way to dissect the function and understand how sensitive it is to changes, which is foundational in the study of dynamic systems.