Composition of functions Discrete Math question

Question

How do I do this? All help is appreciated! Would prefer a step by step tutorial but any help is ok :)

Let $h= g\circ f\circ g$ where $f \colon \mathbb R \to \mathbb Z$ is the floor function and $g \colon \mathbb R\to\mathbb R \colon x\mapsto -x$.
(i) Compute $h(3.4)$, $h(7)$ and $h(-1.3)$.
(ii) Describe what $h$ is doing to a general real number $x$.

Answer 1

Note that:

• $f(x)=\lfloor{x}\rfloor$
• $g(x)=-x$
• $h(x)=g(f(g(x)))=-\lfloor{-x}\rfloor$

Therefore:

• $h( 3.4)=-\lfloor{-3.4}\rfloor=-(-4)= 4$
• $h( 7 )=-\lfloor{-7 }\rfloor=-(-7)= 7$
• $h(-1.3)=-\lfloor{ 1.3}\rfloor=-( 1)=-1$

And as you can probably understand by now, $h(x)=\lceil{x}\rceil$.