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To understand function composition, let's dive into the concept of the outer function. The "outer function" is the part of a composition that handles the final transformation of the variables involved. In simpler terms, think of it like the final step in a process, the part that wraps everything up neatly.

In our exercise example, the outer function is represented as \( g(u) = 5u^3 \). This means that after performing any calculations defined by the inner function (which we'll cover next), the result is then used in this particular outer function. The variable \( u \) stands for whatever comes out of the inner function.

It's helpful to visualize this as a layering process:

• The inner function does its job first,
• then its result becomes the input for the outer function.

This makes the outer function key to understanding the entire expression, as it provides the framework for the final output.

When dealing with function composition, identifying the "inner function" is crucial. This is the function that does the initial transformation to our input variable, setting the stage for the outer function.

In the given exercise, the inner function is expressed as \( h(x) = (x-2) \). Here, \( x \) is manipulated first before anything else occurs. This means the action of "subtracting 2 from \( x \)" is what happens initially. Once this step is complete, the result becomes the input for the outer function.

Understanding the inner function involves recognizing how the initial transformation plays a role in the overall composition.

• First, handle the "inside" operation.
• The outcome then participates further, now as a new piece in the puzzle.

This is akin to setting up the foundations before building the rest of the structure.

Substitution in function composition means placing the result of the inner function into the outer function seamlessly. Picture this as a puzzle: connecting the right pieces to see the whole picture.

In the context of our exercise, substitution happens when we replace \( u \) in the outer function \( g(u) = 5u^3 \) with the expression \( (x-2) \) from the inner function, creating \( y = 5(x-2)^3 \). This is a key step in function composition, ensuring everything fits together.

The process of substitution highlights the interlock between the two functions:

• The inner function's result becomes the placeholder for \( u \) in the outer function.
• This linkage allows us to express one complex operation as an integration of two simpler ones.

Mastering substitution not only simplifies complex expressions into manageable parts but also clarifies how transformations build on each other dynamically.