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When dealing with function composition, the "Inner Function" is a crucial component. This function typically forms the initial part of our composition. It's like laying a foundation.
Your function, let's call it "h", takes the initial input and performs the first operation on it. For example, consider the function composition in part (a) of the given exercise. We have the function \[f(x) = x^2 + 1\]. The inner function here can be chosen as \[h(x) = x^2\], which takes any input value \(x\) and squares it:

• This operation is the first step in processing the input.
• The output of this function, \(x^2\), serves as the input for the next function in the sequence.

So, when building your composed function, identifying the inner function is your starting point, and from there, you can design the outer operation that completes your transformation.

In the world of function composition, once you have identified your inner function, the next step is designing the "Outer Function". This function picks up where the inner function leaves off. Think of it as the finishing touch to our function composition process.
In the exercise given, the outer function is applied to the output of the inner function. For part (a), once we have our input squared by the function \(h(x) = x^2\),

• We then use the outer function \(g(u) = u + 1\).
• Here, \(u\) is the result from the inner function, \(x^2\) in this case.
• The outer function simply adds 1 to this result, completing the operation of \(f(x)\).

Similarly, in part (b), after shifting \(x\) by 3 units with \(h(x) = x-3\),

• The outer function \(g(u) = \frac{1}{u}\) takes over and computes the reciprocal of the outcome, \(x-3\).
• This finalizes our composition in a complete functional transformation of the input.

The outer function is what turns the sequence of operations into the desired result.

A "Reciprocal Function" is a special kind of function that involves flipping a number over 1 (or in simpler terms, dividing 1 by the number).
This transformation is often used to invert values, particularly useful in applications like the exercise's part (b).To express a function such as \(f(x) = \frac{1}{x-3}\) in terms of composition, we utilized a reciprocal step in our outer function.

• Start by taking an input \(x\) and performing the shift operation through the inner function \(h(x) = x - 3\).
• Upon producing the output \(x - 3\), we apply the reciprocal operation through the outer function \(g(u) = \frac{1}{u}\).

This step effectively transforms \(x-3\) into its reciprocal, \(\frac{1}{x-3}\).
Using reciprocal functions is a common and powerful tool in mathematical compositions for functions that need an inverse operation component like division.