Question

Let f : X → Y and g : Y → Z be two invertible functions. Then gof is alsoinvertible with (gof)

–1 = f–1og–1

.​

Answer 1

$\huge\star\mathfrak\blue{{Answer:-}}$

Proof To show that gof is invertible with (gof)

–1 = f–1og–1, it is enough to show that

(f–1og–1)o(gof) = IX and (gof)o(f–1og–1) = IZ

.Now, (f–1og–1)o(gof) = ((f–1og–1) og) of, by Theorem 1

= (f–1o(g–1og)) of, by Theorem 1

= (f–1 o IY) of, by definition of g

–1= IX

.Similarly, it can be shown that (gof ) o (f –1 o g –1) = IZ

.