91Ó°ÊÓ

An "inner function" is the first step in composing two functions. When you look at a function and identify it as a composite (a combination) of functions, the inner function is the one that is applied initially. This function holds the variable expressions that become the input for the next function, known as the outer function.

To illustrate, consider the function from Part (a):

• Function: \[f(x) = \frac{1}{1-x^2} \]

Here, we set our inner function as \(h(x) = x^2\). This choice is made because \(x^2\) lies inside the denominator of the rational function. It's the first modification happening to the input \(x\).In Part (b), the function involves an absolute value:

• Function: \[f(x) = |5 + 2x| \]

The inner function here is chosen to be \(h(x) = 5 + 2x\). The arithmetic inside the absolute value is the inner operation, simplifying it before applying the absolute value as the outer function.

Once the inner function has been dealt with, the "outer function" is the next in line to be applied. The function composition process involves taking the result of the inner function and inserting it into the outer function.For Part (a), after \(h(x) = x^2\) is determined, the outer function needs to take this output and complete the original function:

• Outer Function: \[g(u) = \frac{1}{1-u} \]

This leads us to fulfil the requirement \(g(h(x)) = f(x)\), which effectively reconstructs the original function \(f(x)\) using \(g(u)\).In Part (b), we set \(g(u) = |u|\) as the outer function:

• Outer Function: \[g(u) = |u|\]

By following through with \(g(h(x))\), it ensures that the absolute value operation is the final process applied to \(x\), completing the reconstruction of \(f(x)\). This concept allows for clear understanding and manipulation of composed functions.

The "absolute value function" concerns itself with altering any given real number by removing its sign and considering only its magnitude. This turns any negative input into a positive counterpart, leaving positive inputs unchanged.In Part (b) of the exercise, the expression \(f(x) = |5 + 2x|\) uses an absolute value function to highlight how it impacts the expression it contains. Here, the function \(g(u) = |u|\) is reserved for this purpose. Once we determine that \(h(x) = 5 + 2x\), applying \(g(u)\) finalizes the operation by taking every outcome of \(h(x)\) and converting it into its absolute value, effectively calculating \(f(x)\).Key features of the absolute value function include:

• Non-negative results: The function does not allow negative values in its output.
• Simplicity in computation: Easy application across both algebraic and graphical interpretations.
• Used frequently in equations and inequalities: Offers a strategic technique to simplify expressions involving negative terms.

Understanding the absolute value function's role in compositions is crucial as it sets a defining condition for many mathematical solutions.