Combine functions through composition and "undo" them with inverses.
The composition f∘g (read "f of g") applies g first, then f. The inverse function f⁻¹ reverses the action of f — applying f then f⁻¹ returns you to the original input.
(f∘g)(x)=f(g(x))
f(f−1(x))=xandf−1(f(x))=x
Finding the Inverse Function
1Replace f(x) with y.
2Swap x and y.
3Solve for y.
4Replace y with f⁻¹(x).
5Verify: check that f(f⁻¹(x)) = x.
f(x)=3x−7⟹y=3x−7⟹x=3y−7⟹y=3x+7⟹f−1(x)=3x+7
🔄
The graph of f⁻¹ is the reflection of the graph of f across the line y = x.
💡
Remember This!
A function has an inverse function if and only if it is one-to-one (passes the horizontal line test). Restricted domains can make non-one-to-one functions invertible.