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Let $\mathrm{f}(\mathrm{x})$ be the greatest integer function and $\mathrm{g}(\mathrm{x})$ 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?
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What is $(\mathrm{g} \circ \mathrm{f})\left(-\frac{5}{3}\right)-(\mathrm{f} \circ \mathrm{g})\left(-\frac{5}{3}\right)$ equal to?
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The correct answer is:
1
$f(x) \rightarrow$ greatest integer function
$f(x)=[x]$
$g(x) \rightarrow$ modulus function $g(x)=|x|$
$\begin{aligned} \operatorname{gof}(x)=g(f(x)) & \quad \operatorname{fog}(x)=f(g(x)) \\=g([x]) &=f(|x|) \\=|[x]| &=[|x|] \\ \operatorname{gof}\left(-\frac{5}{3}\right)=\left|\left[-\frac{5}{3}\right]\right| ; f o g\left(-\frac{5}{3}\right)=\left[\left|-\frac{5}{3}\right|\right] \\=|-2| ;=\left[\frac{5}{3}\right] \\=2 ;=1 \\ \operatorname{gof}\left(-\frac{5}{3}\right)-\operatorname{fog}\left(-\frac{5}{3}\right)=2-1=1 \end{aligned}$
$f(x)=[x]$
$g(x) \rightarrow$ modulus function $g(x)=|x|$
$\begin{aligned} \operatorname{gof}(x)=g(f(x)) & \quad \operatorname{fog}(x)=f(g(x)) \\=g([x]) &=f(|x|) \\=|[x]| &=[|x|] \\ \operatorname{gof}\left(-\frac{5}{3}\right)=\left|\left[-\frac{5}{3}\right]\right| ; f o g\left(-\frac{5}{3}\right)=\left[\left|-\frac{5}{3}\right|\right] \\=|-2| ;=\left[\frac{5}{3}\right] \\=2 ;=1 \\ \operatorname{gof}\left(-\frac{5}{3}\right)-\operatorname{fog}\left(-\frac{5}{3}\right)=2-1=1 \end{aligned}$
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