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Function composition is like creating a new function by combining two or more existing functions. Imagine it as fitting one function inside another, similar to a nesting doll. For instance, if you have two functions \( f(x) \) and \( g(x) \), the composition \( f \circ g \) means you first apply \( g \) to \( x \), then take the output of \( g(x) \) and feed it into \( f \).

• Apply the inner function first.
• Use the output as the input for the next function.
• Continue this process until all functions are applied.

In our exercise, three functions are given: \( f(x) = \frac{1}{x} \), \( g(x) = x^3 \), and \( h(x) = x^2 + 2 \). We need to find \( f \circ g \circ h \). Start with \( h(x) \), pass the result to \( g(x) \), then finally to \( f(x) \). Each function affects what the next one does, so stick to the specific order to avoid mistakes.

The order of operations in mathematics ensures that calculations are performed in the correct sequence. Just like how you follow a recipe, using the right order maintains accuracy. For function composition, the sequence in which functions are applied is crucial.

• Start from the innermost function.
• Proceed step-by-step outward.
• Follow the mathematical precedence of operations.

In our composite function \( f \circ g \circ h \), first find \( h(x) \) because it is the innermost. Calculate \( g(h(x)) \) next, as \( g \) is the next layer, and finish with \( f(g(h(x))) \), respecting the given function order. Confusing the order could lead to incorrect results, so it's vital to approach each layer carefully.

Simplifying expressions is about making them easier to read or solve without changing their value. It’s like cleaning up a sentence so it reads better but says the same thing. Often, you can simplify by combining like terms or applying rules.

• Look for like terms.
• Apply distributive, associative, or commutative laws if necessary.
• Simplify fractions, powers, and expressions carefully.

In step 2 of our exercise, we found \( g \circ h(x) = (x^2 + 2)^3 \). Although this can further expand using the binomial theorem, sometimes it's better to keep it unsimplified for clarity and ease of transition to the next steps, unless a further simplification is specifically needed. Simplifying expressions can make upcoming calculations more manageable and help prevent errors.