Relations / Function Composition Inverse
Least You Need to Know: Composition and Inverses
Function composition means applying one function and then another. A function has an inverse only when each output comes from exactly one input.
Least you need to know
(f∘g)(x)meansf(g(x)), so the order matters.- An inverse undoes a function.
- A function must be one-to-one to have an inverse on its codomain.
- Composition can change domain restrictions.
Key notation
- f∘g — apply g, then f
- f^{-1} — inverse function
- f(f^{-1}(y)) — returns y when the inverse exists
Worked example
- Let
f(x)=2x+1andg(x)=x^2. - Then
(f∘g)(3)=f(9)=19. - If
fis one-to-one, its inverse solvesy=2x+1, sof^{-1}(y)=(y-1)/2.
Common mistakes
- Students often compute
g(f(x))when asked forf(g(x)). - Students often think every function has an inverse.
- Students often ignore domain restrictions when taking square roots.
How to recognize it
- Read composition from right to left.
- To find an inverse formula, solve
y=f(x)for x. - If two inputs share one output, an inverse function cannot work.
Next recommended lesson
Continue through this topic with Least You Need to Know: Injective, Surjective, and Bijective Functions.
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