Composition and inverses

Composition means doing one function after another. If f : A → B and g : B → C, then gof : A → C is defined by gof(x) = g(f(x)).

Read it from the inside: apply f first, then g to the result. Example: f(x) = x + 3 and g(x) = x². Then gof(x) = (x + 3)², while fog(x) = x² + 3. These are different, so order matters: in general gof ≠ fog.

A function f is invertible if there is a function g that undoes it: gof is the identity on A and fog is the identity on B. Then g is called f⁻¹.

The key fact: f is invertible exactly when it is a bijection (one-one and onto). A many-one function cannot be undone (which input would you go back to?), and a function that is not onto leaves some outputs with nowhere to come from.

To find the inverse: write y = f(x) and solve for x in terms of y. That expression is f⁻¹(y).

Example: f(x) = (3x – 1)/5 on R. Then y = (3x – 1)/5 gives 5y = 3x – 1, so x = (5y + 1)/3. So f⁻¹(y) = (5y + 1)/3. Check: f((5y + 1)/3) = (5y + 1 – 1)/5 = y. ✓

Think of getting dressed: socks first, then shoes. To undo it, you take off the shoes first, then the socks. In the same way, (gof)⁻¹ = f⁻¹ o g⁻¹: the inverses come in reverse order.