【数学I】例題3.1.2：関数の値域（One More）★

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% 例題I3.1.2：関数の値域（One More）★
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次の関数のグラフをかき，その値域を求めよ．

(1) $y=3x-4(-1\leqq x<2)$

(2) $y=\frac{1}{2}x^2(-2\leqq x\leqq 3)$

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% 例題I3.1.2の解答（One More）★
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(1) $y=3x-4$において，

$x=-1$のとき

\[
y=3\cdot(-1)-4=-7
\]

$x=2$のとき

\[
y=3\cdot 2-4=2
\]

よって，グラフは右の図のようになる．

値域は$-7\leqq y<2$

(2) $y=\frac{1}{2}x^2$において，

$x=-2$のとき

\[
y=\frac{1}{2}\cdot(-2)^2=2
\]

$x=3$のとき

\[
y=\frac{1}{2}\cdot 3^2=\frac{9}{2}
\]

よって，グラフは右の図のようになる．

値域は$0\leqq y\leqq\frac{9}{2}$

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% 問題I3.1.2：関数の値域（One More）★
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次の関数のグラフをかき，その値域を求めよ．

(1) $y=-2x+5(-2\leqq x\leqq 3)$

(2) $y=x^2(-1\leqq x\leqq 2)$

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% 問題I3.1.2の解答（One More）★
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(1) $y=-2x+5$において，

$x=-2$のとき

\[
y=-2\cdot(-2)+5=4+5=9
\]

$x=3$のとき

\[
y=-2\cdot 3+5=-6+5=-1
\]

よって，グラフは右の図のようになる．

値域は$-1\leqq y\leqq 9$

(2) $y=x^2$において，

$x=-1$のとき

\[
y=(-1)^2=1
\]

$x=2$のとき

\[
y=2^2=4
\]

よって，グラフは右の図のようになる．

値域は$0\leqq y\leqq 4$

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% 例題I3.1.2の側注：関数の値域（One More）★
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% 側注1
$y=3x-4$のグラフは，$y$切片が$-4$，傾きが$3$の直線である．

% 側注2
グラフには，定義域の両端の座標を記す．

% 側注3
$y=\frac{1}{2}x^2$のグラフは，頂点が原点，軸が$y$軸であり，下に凸の放物線である．

% 側注4
定義域の端点が値域の端点になるとは限らないので注意すること．

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% 問題I3.1.2の側注：関数の値域（One More）★
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% 側注1
$y=-2x+5$のグラフは，$y$切片が$5$，傾きが$-2$の直線である．

% 側注2
グラフには，定義域の両端の座標を記す．

% 側注3
$y=x^2$のグラフは，頂点が原点，軸が$y$軸であり，下に凸の放物線である．

% 側注4
定義域の端点が値域の端点になるとは限らないので注意すること．

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% 例題I3.1.2の考え方・補足など：関数の値域（One More）★
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% 考え方
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関数$y=f(x)$において，$y$のとる値の範囲を値域という．グラフをかくときは，端点がグラフに含まれるか否かに注意すること．

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% 注意
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数直線のときと同様に，本書ではグラフにおいて端点を含まない場合は「$\circ$」，端点を含む場合は「$\bullet$」で表す．