【数学A】例題4.2.17：ガウス記号を含むグラフ（One More）★★

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% 例題A4.2.17：ガウス記号を含むグラフ（One More）★★
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$[x]$を$x$以下の最大の整数とするとき，次の問いに答えよ．

(1) $[2.7],[-2.5],[4],[\pi+2]$の値を求めよ．

(2) $-3\leqq x\leqq 2$のとき，関数$y=[x]$のグラフをかけ．

(3) $-3\leqq x\leqq 2$のとき，関数$y=x-[x]$のグラフをかけ．

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(1) $2\leqq 2.7<3$であるから，$[2.7]=2$

$-3\leqq-2.5<-2$であるから，$[-2.5]=-3$

$4\leqq 4<5$であるから，$[4]=4$

$5\leqq\pi+2<6$であるから，$[\pi+2]=5$

(2) $-3\leqq x<-2$のとき，$y=[x]=-3$

$-2\leqq x<-1$のとき，$y=[x]=-2$

$-1\leqq x<0$のとき，$y=[x]=-1$

$0\leqq x<1$のとき，$y=[x]=0$

$1\leqq x<2$のとき，$y=[x]=1$

$x=2$のとき，$y=[x]=2$

よって，グラフは右の図のようになる．

(3) $-3\leqq x<-2$のとき，$y=x-[x]=x-(-3)=x+3$

$-2\leqq x<-1$のとき，$y=x-[x]=x-(-2)=x+2$

$-1\leqq x<0$のとき，$y=x-[x]=x-(-1)=x+1$

$0\leqq x<1$のとき，$y=x-[x]=x$

$1\leqq x<2$のとき，$y=x-[x]=x-1$

$x=2$のとき，$y=x-[x]=2-2=0$

よって，グラフは右の図のようになる．

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% 問題A4.2.17：ガウス記号を含むグラフ（One More）★★
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$[x]$を$x$以下の最大の整数とするとき，次の問いに答えよ．

(1) $[3.2],[-0.7],[2],[\sqrt{7}+1]$の値を求めよ．

(2) $-1\leqq x\leqq 2$のとき，関数$y=2[x]$のグラフをかけ．

(3) $-1\leqq x\leqq 2$のとき，関数$y=-[2x]$のグラフをかけ．

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(1) ${3}\leqq 3.2<4$であるから，$[3.2]=3$

${-1}\leqq-0.7<0$であるから，$[-0.7]=-1$

${2}\leqq 2<3$であるから，$[2]=2$

${3}\leqq\sqrt{7}+1<4$であるから，$[\sqrt{7}+1]=3$

(2) ${-1}\leqq x<0$のとき，$y=2[x]=-2$

${0}\leqq x<1$のとき，$y=2[x]=0$

${1}\leqq x<2$のとき，$y=2[x]=2$

$x=2$のとき，$y=2[x]=4$

よって，グラフは右の図のようになる．

(3) $-1\leqq x<-\frac{1}{2}$のとき，$y=-[2x]=2$

$-\frac{1}{2}\leqq x<0$のとき，$y=-[2x]=1$

$0\leqq x<\frac{1}{2}$のとき，$y=-[2x]=0$

$\frac{1}{2}\leqq x<1$のとき，$y=-[2x]=-1$

$1\leqq x<\frac{3}{2}$のとき，$y=-[2x]=-2$

$\frac{3}{2}\leqq x<2$のとき，$y=-[2x]=-3$

$x=2$のとき，$y=-[2x]=-4$

よって，グラフは右の図のようになる．

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% 例題A4.2.17の側注：ガウス記号を含むグラフ（One More）★★
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% 側注1
$[-2.5]=-2$ではないので注意すること．

% 側注2
円周率$\pi$は$3\leqq\pi<4$を満たすので，$5\leqq\pi+2<6$

% 側注3
$-3\leqq x<-2$のとき，
\[
[x]=-3
\]

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% 問題A4.2.17の側注：ガウス記号を含むグラフ（One More）★★
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% 側注1
$[-0.7]=0$ではないので注意すること．

% 側注2
$2\leqq\sqrt{7}<3$より，$3\leqq\sqrt{7}+1<4$

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% 例題A4.2.17の考え方・補足など：ガウス記号を含むグラフ（One More）★★
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実数$x$について，$x$以下の最大の整数を$[x]$と表す（[]を{ガウス記号}という）．また，$n$を整数とするとき，$n\leqq x<n+1$ならば$[x]=n$が成り立つことを利用して，場合分けをする．

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実数$x$について，$n$を整数とするとき，$n\leqq x<n+1\Longleftrightarrow[x]=n$