関数の定義(definition of functions)

確認問題

1.

次の関数の定義域と値域を求めよう．

(a) $\displaystyle{f(x,y) = \sqrt{xy}}$ (b) $\displaystyle{f(x,y) = \frac{1}{x+y}}$ (c) $\displaystyle{f(x,y) = \frac{1}{x^{2} + y^{2}}}$

(d) $\displaystyle{f(x,y) = \frac{x^{2}}{x^{2} + y^{2}}}$ (e) $\displaystyle{f(x,y) = \log(1-xy)}$ (f) $\displaystyle{f(x,y,z) = \frac{z}{x^{2} - y^{2}}}$

2.

次の曲面を分類しよう．

(a) $\displaystyle{x^{2} + 4y^{2} - 16z^{2} = 0}$ (b) $\displaystyle{x^{2} + 4y^{2} + 16z^{2}- 12 = 0}$ (c) $\displaystyle{x - 4y^{2} = 0}$

(d) $\displaystyle{x^{2} - 4y^{2} - 2z = 0}$ (e) $\displaystyle{2x^{2} + 4y^{2} - 1 = 0}$ (f) $\displaystyle{x^{2} + 4y^{2} - 4z = 0}$

(g) $\displaystyle{2x^{2} - 4y^{2} - 6 = 0}$ (h) $\displaystyle{x^{2} + y^{2} - 2z^{2} -10 = 0}$ (i) $\displaystyle{x^{2} + y^{2} - 2z^{2} +10 = 0}$

演習問題

1.: 次の関数の定義域をもとめそのグラフを描こう．
(a) $\displaystyle{f(x,y) = x^2 - y^2}$ (b) $\displaystyle{f(x,y) = \frac{x^2}{x^2 + y^2}}$ (c) $\displaystyle{f(x,y) = \log{(1 - xy)}}$