{"id":248,"date":"2020-07-04T21:55:05","date_gmt":"2020-07-04T12:55:05","guid":{"rendered":"https:\/\/tako.nakano.net\/blog\/?p=248"},"modified":"2020-07-07T22:17:42","modified_gmt":"2020-07-07T13:17:42","slug":"fx_eq_2f2x","status":"publish","type":"post","link":"https:\/\/tako.nakano.net\/blog\/2020\/07\/fx_eq_2f2x\/","title":{"rendered":"f(x)=2 f(2x) \u3092\u6e80\u305f\u3059\u95a2\u6570\u306b\u3064\u3044\u3066"},"content":{"rendered":"

\u554f\u984c<\/h1>\n

\u95a2\u6570\u65b9\u7a0b\u5f0f\"f(2x)=\frac\u3092\u6e80\u305f\u3059\u95a2\u6570f(x)\u3092\u6c42\u3081\u3088\uff0e<\/p>\n

\u767a\u898b\u7684\u89e3\u6cd5<\/h2>\n

\u8003\u5bdf1<\/h3>\n

\"x\neq0\"\u3067\u3042\u308b\u3068\u304d\u3092\u8003\u3048\u308b\uff0e\"f(x)=\frac\u3068\u3059\u308b\u3068\uff0c\"f(2x)=\frac\u3088\u308a\uff0c(1)\u3092\u6e80\u305f\u3059\uff0e<\/p>\n

\u5168\u3066\u306e\u5b9f\u6570\"x\"\u3067\u5b9a\u7fa9\u3055\u308c\u3066\u3044\u308b\u3053\u3068\u304c\u5fc5\u8981\u3067\u3042\u308c\u3070\uff0c\"f(0)=0\"\u3068\u5b9a\u7fa9\u3059\u308c\u3070\u3088\u3044\uff0e\"f(2\cdot0)=\frac\u3067\u3042\u308b\u304b\u3089\uff0c\"f(0)=0\"\u3067\u3042\u308b\u3053\u3068\u304c\u3059\u3050\u5206\u304b\u308b\uff0e<\/p>\n

\u8003\u5bdf2<\/h3>\n

\u540c\u69d8\u306b\u8003\u3048\u3066\uff0c\"f(x)=\begin{cases}c\cdot\frac\uff08\"c\"\u306f\u4efb\u610f\u306e\u5b9a\u6570\uff09\u3068\u5b9a\u7fa9\u3059\u308b\u3068\uff0c\u3053\u306e\u3088\u3046\u306a\u95a2\u6570\u306f\u6761\u4ef6(1)\u3092\u6e80\u305f\u3059\uff0e<\/p>\n

\u8003\u5bdf3<\/h3>\n

\u3067\u306f\uff0c\u3053\u306e\u3088\u3046\u306a\u95a2\u6570\u306e\u307f\u304c\u6761\u4ef6\u3092\u6e80\u305f\u3059\u304b\u3068\u3044\u3046\u3068\uff0c\u305d\u306e\u3088\u3046\u306a\u3053\u3068\u306f\u306a\u3044\uff0e\u4f8b\u3048\u3070\"f(1)=1\"\u3068\u5b9a\u3081\u305f\u3068\u3057\u3066\uff0c\u5f71\u97ff\u3092\u53d7\u3051\u308b\u3088\u3046\u306a\"x\"\u306f\u3069\u306e\u3088\u3046\u306a\"x\"\u304b\u8003\u3048\u3066\u307f\u308b\uff0e<\/p>\n

\"x=2^{n}\"\uff08\"n\"\u306f\u6574\u6570\uff09\u3092\u6e80\u305f\u3059\u3088\u3046\u306a\"x\"\u306b\u3064\u3044\u3066\u306f\uff0c\"f(2^{n})=\frac\u304c\u76f4\u3061\u306b\u5206\u304b\u308b\uff0e\u9006\u306b\uff0c\u305d\u308c\u4ee5\u5916\u306e\"x\"\u306e\u5024\u306f\u5b9a\u3081\u306a\u3044\uff0e<\/p>\n

\u3088\u3063\u3066\uff0c\u6b21\u306e\u95a2\u6570\u306f\u6761\u4ef6\u3092\u6e80\u305f\u3059\uff0e\"f(x)=\begin{cases}-\frac\u30b0\u30e9\u30d5\u3092\u304b\u304f\u3068\uff0c\u3068\u3053\u308d\u3069\u3053\u308d\u3067\u7a74\u304c\u958b\u3044\u305f\u3088\u3046\u306a\u30b0\u30e9\u30d5\u306b\u306a\u308b\uff0e<\/p>\n

\u8003\u5bdf4<\/h3>\n

\u5225\u306e\u95a2\u6570\u65b9\u7a0b\u5f0f\"f(x)=f(2x)\quad(2)\"\u3092\u6e80\u305f\u3059\u95a2\u6570\u306e\u5834\u5408\uff0chttps:\/\/detail.chiebukuro.yahoo.co.jp\/qa\/question_detail\/q1487297239\uff0chttps:\/\/oshiete.goo.ne.jp\/qa\/8700628.html\u7b49\u304b\u3089\u4ee5\u4e0b\u306e\u3053\u3068\u304c\u5206\u304b\u308b\uff0e<\/p>\n

\u5b9f\u6570\u5168\u4f53\u3067\u5b9a\u7fa9\u3055\u308c\u308b\u9023\u7d9a\u95a2\u6570\u3067\u3042\u308c\u3070\uff0c\"f(x)=c\"\uff08\"c\"\u306f\u4efb\u610f\u306e\u5b9a\u6570\uff09\u3067\u3042\u308b\uff0e<\/p>\n

\"x>0\"\u3067\u5b9a\u7fa9\u3055\u308c\u308b\u9023\u7d9a\u95a2\u6570\u3067\u3042\u308b\u3068\u3059\u308b\uff0e\"g(x)=2\pi\cdot\log_{2}x\"\u3068\u3059\u308b\u3068\uff0c\"g(2x)=g(x)+2\pi\"\u3092\u6e80\u305f\u3059\uff0e\u3088\u3063\u3066\uff0c\u95a2\u6570\"f(x)=\sin\left(g(x)\right)\"\u306f\u6761\u4ef6(2)\u3092\u6e80\u305f\u3059\uff0e\u3064\u307e\u308a\uff0c\u5b9a\u6570\u95a2\u6570\u3067\u306f\u306a\u3044\u304c\uff0c\u6761\u4ef6\u3092\u6e80\u305f\u3059\u95a2\u6570\u306f\u3044\u304f\u3089\u3067\u3082\u69cb\u6210\u3067\u304d\u308b\uff0e<\/p>\n

\u8003\u5bdf5<\/h3>\n

\u95a2\u6570\u65b9\u7a0b\u5f0f(1)\u3092\u6e80\u305f\u3059\uff0c\"x>0\"\u3067\u5b9a\u7fa9\u3055\u308c\u308b\u9023\u7d9a\u95a2\u6570\u306b\u3064\u3044\u3066\u518d\u5ea6\u8003\u5bdf\u3059\u308b\uff0e\u524d\u9805\u3092\u5229\u7528\u3059\u308b\u3068\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u89e3\u304c\u69cb\u6210\u3067\u304d\u308b\uff0e\"g_{2\pi}(x)=2\pi\cdot\log_{2}x\"\u3068\u3059\u308b\u3068\uff0c\"g_{2\pi}(2x)=g_{2\pi}(x)+2\pi\"\u3092\u6e80\u305f\u3059\uff0e\u95a2\u6570\"f(x)=\frac\u306f\u6761\u4ef6(1)\u3092\u6e80\u305f\u3059\uff0e<\/p>\n

\"x<0\"\u306b\u3064\u3044\u3066\u3082\uff0c\u3053\u306e\u7bc4\u56f2\u3067\u9023\u7d9a\u306a\u95a2\u6570\u304c\"f(x)=\frac\u3084\"f(x)=\frac\u306e\u3088\u3046\u306b\u3059\u308c\u3070\u69cb\u6210\u3067\u304d\u308b\uff0e<\/p>\n

\u307e\u3068\u3081<\/h3>\n

\u95a2\u6570\u65b9\u7a0b\u5f0f\"f(2x)=\frac\u306b\u3064\u3044\u3066\uff0c\u4ee5\u4e0b\u306e\u3053\u3068\u304c\u5206\u304b\u308b\uff0e<\/p>\n

    \n
  1. \u95a2\u6570\"f(x)=0\"\u306f\uff0c\u3059\u3079\u3066\u306e\u5b9f\u6570\u3067\u5b9a\u7fa9\u3055\u308c\u308b\u9023\u7d9a\u95a2\u6570\u3067\uff0c\u6761\u4ef6\u3092\u6e80\u305f\u3059\uff0e<\/li>\n
  2. \u95a2\u6570\"f(x)=\begin{cases}c\cdot\frac\uff08\"c\"\u306f\u4efb\u610f\u306e\u5b9a\u6570\uff09\u306f\uff0c\"x\neq0\"\u3067\u9023\u7d9a\u3067\uff0c\u6761\u4ef6\u3092\u6e80\u305f\u3059\uff0e<\/li>\n
  3. \u95a2\u6570\"f(x)=\begin{cases}c\cdot\frac\uff08\"c\"\u306f\u4efb\u610f\u306e\u5b9a\u6570\uff09\u306f\uff0c\"x\neq0\"\u3067\u9023\u7d9a\u3067\uff0c\u6761\u4ef6\u3092\u6e80\u305f\u3059\uff0e<\/li>\n
  4. \u4e0d\u9023\u7d9a\u306a\u95a2\u6570\u3067\u3042\u308c\u3070\uff0c\u3044\u304f\u3089\u3067\u3082\u898b\u3064\u304b\u308b\uff0e<\/li>\n<\/ol>\n

    \u5206\u304b\u3063\u3066\u3044\u306a\u3044\u3053\u3068\u306b\uff0c\u4f8b\u3048\u3070\u4ee5\u4e0b\u306e\u3053\u3068\u304c\u3042\u308b\uff0e<\/p>\n

      \n
    1. \"x=0\"\u3067\u9023\u7d9a\u306a\u95a2\u6570\u3067\uff0c\u5b9a\u6570\u95a2\u6570\"f(x)=0\"\u3067\u306a\u3044\u95a2\u6570\u306f\u89e3\u3068\u3057\u3066\u5b58\u5728\u3059\u308b\u304b\uff0e<\/li>\n<\/ol>\n

      \u69cb\u6210\u7684\u89e3\u6cd5<\/h2>\n

      \u5168\u3066\u306e\u5b9f\u6570\u3067\u9023\u7d9a<\/h3>\n

      \"f(x)\"\u304c\u5168\u3066\u306e\u5b9f\u6570\u3067\u5b9a\u7fa9\u3055\u308c\uff0c\u9023\u7d9a\u3067\u3042\u308b\u3068\u3057\uff0c\u95a2\u6570\u65b9\u7a0b\u5f0f\u306e\u89e3\u306b\u3064\u3044\u3066\u8003\u5bdf\u3059\u308b\uff0e<\/p>\n

      \"f(2x)=\frac\u3088\u308a\uff0c\u4efb\u610f\u306e\u81ea\u7136\u6570\"n\"\u306b\u5bfe\u3057\u3066\uff0c\"f(2^{n}x)=\frac\u3067\u3042\u308b\u3053\u3068\u304c\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u3092\u7528\u3044\u3066\u8a3c\u660e\u3067\u304d\u308b\uff0e\u3088\u3063\u3066\uff0c\u4efb\u610f\u306e\u6574\u6570\"n\"\u306b\u5bfe\u3057\u3066\uff0c\"f(2^{n}x)=2^{-n}\cdot\u3067\u3042\u308b\u3053\u3068\u3082\u8a3c\u660e\u3067\u304d\u308b\uff0e<\/p>\n

      \"f(0)=c\"\u3068\u3059\u308b\uff0e<\/p>\n

      \u95a2\u6570\u306f\"x=0\"\u3067\u9023\u7d9a\u3067\u3042\u308b\u304b\u3089\uff0c\u4efb\u610f\u306e\u5b9f\u6570\"\varepsilon>0\"\u306b\u5bfe\u3057\uff0c\u3042\u308b\u5b9f\u6570\"\delta>0\"\u304c\u5b58\u5728\u3057\uff0c\"\left|x-0\right|<\delta\"\u3092\u6e80\u305f\u3059\u4efb\u610f\u306e\u5b9f\u6570\"x\"\u306b\u3064\u3044\u3066\uff0c\"\left|f(x)-c\right|<\varepsilon\"\u3067\u3042\u308b\uff0e\u4ee5\u4e0b\uff0c\"\varepsilon\"\uff0c\"\delta\"\u306f\u56fa\u5b9a\u3059\u308b\uff0e<\/p>\n

      \"\left|x_{0}-0\right|<\delta\"\u3092\u6e80\u305f\u3059\uff0c\u3042\u308b\u5b9f\u6570\"x_{0}\"\u3092\u56fa\u5b9a\u3059\u308b\uff0e\u81ea\u7136\u6570\"N\"\u306b\u5bfe\u3057\u3066\uff0c\"x_{N}=\frac\u3068\u5b9a\u3081\u308b\u3068\uff0c\"\left|x_{N}-0\right|=\left|\frac\u3067\u3042\u308b\uff0e\u307e\u305f\uff0c\"f(x_{N})=f\left(\frac\u3067\u3042\u308b\u304b\u3089\uff0c\"\left|f(x_{N})-c\right|=\left|2^{N}f(x_{0})-c\right|\"\u3067\u3042\u308b\uff0e\\\\
      \n(i) \"f(x_{0})=0\"\u306e\u3068\u304d\uff0c\"\left|c\right|<\varepsilon\"\u3067\u3042\u308b\u304b\u3089\uff0c\"c=0\"\u3067\u3042\u308b\uff0e\u3053\u306e\u3068\u304d\"\left|2^{N}f(x_{0})-c\right|=0<\varepsilon\"\u3067\u3042\u308a\uff0c\u6761\u4ef6\u3092\u6e80\u305f\u3059\uff0e\\\\
      \n(ii) \"f(x_{0})\neq0\"\u306e\u3068\u304d\uff0c\"\left|2^{N}f(x_{0})-c\right|\geqq\left|2^{N}f(x_{0})\right|-\left|c\right|=2^{N}\left|f(x_{0})\right|-\left|c\right|\"\u3067\u3042\u308b\uff0e\"\left|f(x_{0})\right|>0\"\u3067\u3042\u308b\u304b\u3089\uff0c\"\varepsilon_{1}=\varepsilon+\left|c\right|>0\"\u306b\u3064\u3044\u3066\uff0c\"2^{N_{1}}\left|f(x_{0})\right|>\varepsilon_{1}\"\u3092\u6e80\u305f\u3059\u3088\u3046\u306a\u81ea\u7136\u6570\"N_{1}\"\u304c\u5b58\u5728\u3059\u308b\uff0e\u3053\u306e\u3088\u3046\u306a\u81ea\u7136\u6570\"N_{1}\"\u306b\u5bfe\u3057\u3066\uff0c\"2^{N_{1}}\left|f(x_{0})\right|-\left|c\right|>\varepsilon+\left|c\right|-\left|c\right|>\varepsilon\"\u3067\u3042\u308b\uff0e\u3064\u307e\u308a\uff0c\"\left|f(x_{N_{1}})-c\right|>\varepsilon\"\u3067\u3042\u308b\uff0e\u3053\u308c\u306f\uff0c\"f(x)\"\u306e\u9023\u7d9a\u6027\u306b\u77db\u76fe\u3059\u308b\uff0e\uff08\u80cc\u7406\u6cd5\u306f\u5acc\u3044\u3060\u3051\u3069\u7528\u3044\u305f\uff09
      \n\u5f93\u3063\u3066\uff0c\"f(x_{0})=0\"\u306e\u307f\u9069\u3059\u308b\uff0e<\/p>\n

      \u4ee5\u4e0a\u306e\u8b70\u8ad6\u306f\"\left|x_{0}-0\right|<\delta\" \u3092\u6e80\u305f\u3059\u4efb\u610f\u306e\u5b9f\u6570\"x_{0}\"\u306b\u3064\u3044\u3066\u6210\u308a\u7acb\u3064\uff0e\u3088\u3063\u3066\uff0c\"\left|x-0\right|<\delta\"\u306a\u3089\u3070\"f(x)=0\"\u3067\u3042\u308b\uff0e<\/p>\n

      \u3055\u3066\uff0c\"\left|x_{2}-0\right|<\delta\"\u3068\u306f\u9650\u3089\u306a\u3044\uff0c\u4efb\u610f\u306e\u5b9f\u6570\"x_{2}\"\u3092\u5225\u306b\u3068\u308b\uff0e\u4efb\u610f\u306e\u5b9f\u6570\"x_{2}\"\u306b\u5bfe\u3057\uff0c\"x_{3}=\frac\u304c\"\left|x_{3}-0\right|<\delta\"\u3092\u6e80\u305f\u3059\u3088\u3046\u306a\u81ea\u7136\u6570\"N_{2}\"\u304c\u5b58\u5728\u3059\u308b\uff0e\u3053\u306e\u3068\u304d\uff0c\"f(x_{3})=0\"\u3067\u3042\u308b\uff0e<\/p>\n

      \u3088\u3063\u3066\uff0c\"f(x_{2})=f\left(2^{N_{2}}x_{3}\right)=2^{-N_{2}}\cdot\u3067\u3042\u308b\u304b\u3089\uff0c\"f(x_{2})=0\"\u3067\u3042\u308b\uff0e<\/p>\n

      \u4ee5\u4e0a\u3088\u308a\uff0c\u4efb\u610f\u306e\u5b9f\u6570\"x\"\u306b\u5bfe\u3057\u3066\"f(x)=0\"\u3067\u3042\u308b\uff0e<\/p>\n

      \"\boldsymbol{x\neq0}\"\u3067\u9023\u7d9a<\/h3>\n

      \"x>0\"\u3068\"x<0\"\u306b\u3064\u3044\u3066\u5225\u3005\u306b\u8003\u3048\u3066\u3088\u304f\uff0c\u5bfe\u79f0\u6027\u304b\u3089\"x>0\"\u306e\u3068\u304d\u3060\u3051\u3092\u8003\u3048\u308c\u3070\u5341\u5206\u3067\u3042\u308b\uff0e<\/p>\n

      \"x>0\"\u3067\u5b9a\u7fa9\u3055\u308c\u308b\u5468\u671f1\u306e\u9023\u7d9a\u95a2\u6570\u3092\u4efb\u610f\u306b\u5b9a\u3081\uff0c\"h(x)\"\u3068\u3059\u308b\uff0e\u3064\u307e\u308a\uff0c\"h(x+1)=h(x)\"\u3092\u6e80\u305f\u3059\uff0e\u307e\u305f\uff0c\u95a2\u6570\"g(x)\"\u3092\"g(x)=\log_{2}x\"\u3068\u5b9a\u3081\u308b\uff0e\u3053\u306e\u3068\u304d\uff0c\u4efb\u610f\u306e\u81ea\u7136\u6570\"N\"\u306b\u5bfe\u3057\uff0c\"h\left(g\left(2x\right)\right)=h\left(1+\log_{2}x\right)=h(\log_{2}x)=h\left(g\left(x\right)\right)\"\u3067\u3042\u308b\uff0e\"h\left(g\left(x\right)\right)\"\u306f\"x>0\"\u3067\u9023\u7d9a\u3067\u3042\u308b\uff08\u8a3c\u660e\u7565\uff09\uff0e<\/p>\n

      \u3053\u306e\u3068\u304d\uff0c\u95a2\u6570\"f_{1}(x)\"\u3092\"f_{1}(x)=\frac\u3068\u5b9a\u3081\u308b\uff0e\u95a2\u6570\"f_{1}(x)\"\u306f\"f_{1}(2x)=\frac\u3067\u3042\u308b\u306e\u3067\uff0c\u95a2\u6570\u65b9\u7a0b\u5f0f(1)\u3092\u6e80\u305f\u3059\uff0e<\/p>\n

      \u3088\u3063\u3066\uff0c\u95a2\u6570\u65b9\u7a0b\u5f0f\u3092\u6e80\u305f\u3059\u95a2\u6570\u304c\uff0c\u4efb\u610f\u306e\u5468\u671f1\u306e\u5468\u671f\u95a2\u6570\u306b\u5bfe\u3057\u3066\uff0c\u305d\u308c\u305e\u308c\u7570\u306a\u308b\u3088\u3046\u306b\u69cb\u6210\u3067\u304d\u308b\uff0e<\/p>\n","protected":false},"excerpt":{"rendered":"

      \u554f\u984c \u95a2\u6570\u65b9\u7a0b\u5f0f\u3092\u6e80\u305f\u3059\u95a2\u6570f(x)\u3092\u6c42\u3081\u3088\uff0e \u767a\u898b\u7684\u89e3\u6cd5 \u8003\u5bdf1 \u3067\u3042\u308b\u3068\u304d\u3092\u8003\u3048\u308b\uff0e\u3068\u3059\u308b\u3068\uff0c\u3088\u308a\uff0c(1)\u3092\u6e80\u305f\u3059\uff0e \u5168\u3066\u306e\u5b9f\u6570\u3067\u5b9a\u7fa9\u3055\u308c\u3066\u3044\u308b\u3053\u3068\u304c\u5fc5\u8981\u3067\u3042\u308c\u3070\uff0c\u3068\u5b9a\u7fa9\u3059\u308c\u3070\u3088\u3044\uff0e\u3067\u3042\u308b\u304b\u3089\uff0c\u3067\u3042\u308b\u3053\u3068\u304c\u3059\u3050\u5206\u304b\u308b […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[3],"tags":[],"class_list":["post-248","post","type-post","status-publish","format-standard","hentry","category-misc"],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4dIdP-40","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/tako.nakano.net\/blog\/wp-json\/wp\/v2\/posts\/248","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/tako.nakano.net\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/tako.nakano.net\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/tako.nakano.net\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/tako.nakano.net\/blog\/wp-json\/wp\/v2\/comments?post=248"}],"version-history":[{"count":5,"href":"https:\/\/tako.nakano.net\/blog\/wp-json\/wp\/v2\/posts\/248\/revisions"}],"predecessor-version":[{"id":255,"href":"https:\/\/tako.nakano.net\/blog\/wp-json\/wp\/v2\/posts\/248\/revisions\/255"}],"wp:attachment":[{"href":"https:\/\/tako.nakano.net\/blog\/wp-json\/wp\/v2\/media?parent=248"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/tako.nakano.net\/blog\/wp-json\/wp\/v2\/categories?post=248"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/tako.nakano.net\/blog\/wp-json\/wp\/v2\/tags?post=248"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}