Q220 / 270 · NDA 2016 (Phase 1) · Mathematics

Let $f(x)$ be the greatest integer function and $g(x)$ be the modulus function. What is $(\mathrm{g} \circ \mathrm{f})\left(-\frac{5}{3}\right)-(\mathrm{f} \circ \mathrm{g})\left(-\frac{5}{3}\right)$ equal to?

  • A
    –1
  • B
    0
  • Correct answer
    C
    1
  • D
    2

Solution

$f(x) \rightarrow$ greatest integer function $f(x)=[x]$ $g(x) \rightarrow$ modulus function $g(x)=|x|$ $\begin{aligned} & g \circ f(x)=g(f(x)) \quad f o g(x)=f(g(x)) \\ & =g([x]) \quad \quad \quad \quad \quad=f(|x|) \\ & =|[x]| \quad \quad \quad \quad \quad=[|x|] \\ & \operatorname{gof}\left(-\frac{5}{3}\right)=\left|\left[-\frac{5}{3}\right]\right| ; f o g\left(-\frac{5}{3}\right)=\left[\left|-\frac{5}{3}\right|\right] \\ & =|-2| \quad ; \quad=\left[\frac{5}{3}\right] \\ & =2 \quad ; \quad=1 \\ & \operatorname{gof}\left(-\frac{5}{3}\right)-f o g\left(-\frac{5}{3}\right)=2-1=1 \end{aligned}$

Hence, the correct answer is option: 'C'.

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