Form and use composite functions, understanding that the order of composition is important

Functions – Composite Functions

What is a composite function?

If we have two functions f(x) and g(x), the composite function f∘g is defined as: $$f∘g(x) = f\bigl(g(x)\bigr).$$ Think of it like a two‑step recipe: first you bake the cake (apply g), then you frost it (apply f). The final taste depends on the order! 🍰

How to form a composite function

1. Write down the two functions. Example: f(x)=2x+3 and g(x)=x^2-1.
2. Replace every occurrence of x in f(x) with the entire expression of g(x):
   f∘g(x)=2\bigl(x^2-1\bigr)+3 = 2x^2-2+3 = 2x^2+1.
3. Check your work by plugging a value, e.g., x=2:
   g(2)=2^2-1=3, then f(3)=2·3+3=9. The composite gives 9 as well.

Order matters! Why does f∘g ≠ g∘f?

Using the same functions:
f∘g(x)=2x^2+1 (as above).
g∘f(x)=\bigl(2x+3\bigr)^2-1 = 4x^2+12x+8.
They are clearly different. The order changes the “ingredients” that each function receives, just like swapping the order of adding sugar and flour changes the cake’s texture. 🔄

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