Q.Let $f, g : \mathbb{R} \to \mathbb{R}$ be defined by $f(x) = |x|$ and $g(x) = [x]$ (the greatest integer function). Examine whether $f$ and $g$ are one-one and onto. Hence find $(f \circ g)(1.5)$ and $(g \circ f)(-2.5)$.

$f(x)=|x|$: not one-one, since $f(1)=1=f(-1)$ but $1\neq-1$. Not onto $\mathbb{R}$, since the range is $[0,\infty)$ and negative values (e.g. $-1$) are never attained.

$g(x)=[x]$ (greatest integer): not one-one, since $g(1.2)=1=g(1.7)$ but $1.2\neq1.7$. Not onto $\mathbb{R}$, since the range is $\mathbb{Z}$ and non-integer values (e.g. $0.5$) are never attained.

Computing $(f\circ g)(1.5)$: $g(1.5)=[1.5]=1$ (the greatest integer not exceeding $1.5$), so $(f\circ g)(1.5)=f(1)=|1|=1$. …

Examine each function's injectivity and surjectivity separately using a concrete counterexample pair and a concrete unattained value, t …

• Evaluating $g(1.5)$ incorrectly as $2$ by rounding instead of taking the greatest integer NOT exceeding $1.5$ (which is $1$). …