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.
The least you need to know
- `(f∘g)(x)` means `f(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
Tiny worked example
- Let `f(x)=2x+1` and `g(x)=x^2`.
- Then `(f∘g)(3)=f(9)=19`.
- If `f` is one-to-one, its inverse solves `y=2x+1`, so `f^{-1}(y)=(y-1)/2`.
Common mistakes
- Students often compute `g(f(x))` when asked for `f(g(x))`.
- Students often think every function has an inverse.
- Students often ignore domain restrictions when taking square roots.
How to recognize this kind of problem
- 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.