Так как по определению f(-x) = -f(x) и
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(-x) = -
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(x), то
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(-x) = f(-x) +
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(-x) = -f(x) -
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(x) = - (f(x) +
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(x)) = -
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(x)
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(-x) = f(-x) -
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(-x) = -f(x) +
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(x) = - (f(x) -
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(x)) = -
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(x).
Полученные равенства означают, что
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(x) и
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(x) – нечётные функции.
Свойство 2. Если y = f(x) и y =
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(x) – нечётные функции, то их произведение и частное есть функция чётная.
Доказательство
Пусть функции y = f(x) и y =
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(x) имеют область определения X, симметричную относительно нуля. Обозначим произведение и частное данных функций Ф
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(x) и Ф
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(x) соответственно:
Ф
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(x) = f(x)
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(x); Ф
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(x) =
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(
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(x)
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0).
Учитывая, что функции f(x) и
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(x) – нечётные, будем иметь:
Ф
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(-x) = f(-x)
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(-x) = (-f(x)) (-
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(x)) = f(x)
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(x) = Ф
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(x);
Ф
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(-x) =
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=
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=
![](data:image/gif;base64,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)
= Ф
![](data:image/gif;base64,R0lGODlhCwAXAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMADAAFAAgAhAAAAAAAAAAzWh1GbDMAADIAMjNbgFozHVtISF1/f0huf2xGHX9ZM25GRn9/XWaIiIBbMwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwUSICCORDAAjViMwMG+xiiMQQ2EADs=)
(x).
Полученные равенства доказывают, что Ф
![](data:image/gif;base64,R0lGODlhCAAXAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIADQAFAAcAgwAAAAAAAAAAMwAcSEgcAEceM1ozAF1/f2aIiIiIZgECAwECAwECAwECAwECAwECAwQOEBABKig2V6wtB0YwABEAOw==)
(x) и Ф
![](data:image/gif;base64,R0lGODlhCwAXAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMADAAFAAgAhAAAAAAAAAAzWh1GbDMAADIAMjNbgFozHVtISF1/f0huf2xGHX9ZM25GRn9/XWaIiIBbMwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwUSICCORDAAjViMwMG+xiiMQQ2EADs=)
(x) функции чётные.
Свойство 3. Если y = f(x) и y =
![](data:image/gif;base64,R0lGODlhDwARAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgALAAsAhAAAAAAAAB0AAB0AHQAAMx0dSB0zWgAzWh1GbDMAADIAMjNbgEgcAEceM1ozAF1/f0huf2xGHX9uSH9/XWaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwUwICBKAgIEwSKKhFgAUQtUcrQOImMCzSrjIsNqB7SxVonVUSlABXZKgCMaVVCVh2sIADs=)
(x) – чётные функции , то их сумма, разность, произведение и частное есть функция чётная.
Пусть функции y = f(x) и y =
![](data:image/gif;base64,R0lGODlhDwARAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgALAAsAhAAAAAAAAB0AAB0AHQAAMx0dSB0zWgAzWh1GbDMAADIAMjNbgEgcAEceM1ozAF1/f0huf2xGHX9uSH9/XWaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwUwICBKAgIEwSKKhFgAUQtUcrQOImMCzSrjIsNqB7SxVonVUSlABXZKgCMaVVCVh2sIADs=)
(x) имеют область определения X, симметричную относительно нуля. Обозначим сумму данных функций G
![](data:image/gif;base64,R0lGODlhCAAXAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIADQAFAAcAgwAAAAAAAAAAMwAcSEgcAEceM1ozAF1/f2aIiIiIZgECAwECAwECAwECAwECAwECAwQOEBABKig2V6wtB0YwABEAOw==)
(x),
разность функций G
![](data:image/gif;base64,R0lGODlhCwAXAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMADAAFAAgAhAAAAAAAAAAzWh1GbDMAADIAMjNbgFozHVtISF1/f0huf2xGHX9ZM25GRn9/XWaIiIBbMwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwUSICCORDAAjViMwMG+xiiMQQ2EADs=)
(x), произведение функций G
![](data:image/gif;base64,R0lGODlhCQAYAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIADAAFAAgAhAAAAAAAAB0AAAAzWh1GbDMeRzNbgEceM1szAFtISEhZf11/f39uSH9/XXtqRGaIiIBuboiIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwUTICCOQkCMQIEqIhKgwAGLQWAAIQA7)
(x), частное данных функций G
![](data:image/gif;base64,R0lGODlhCwAXAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIADAAGAAgAgwAAAAAAAB0zWgAzWgAzWx0zWzMAADNZf0gdAFszM1tuSHtqRH9/XX9uSIBbM4iIZgQSEMg5FwVmUHWBkEggEtQ3NVIEADs=)
(x) соответственно:
G
![](data:image/gif;base64,R0lGODlhCAAXAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIADQAFAAcAgwAAAAAAAAAAMwAcSEgcAEceM1ozAF1/f2aIiIiIZgECAwECAwECAwECAwECAwECAwQOEBABKig2V6wtB0YwABEAOw==)
(x) = f(x) +
![](data:image/gif;base64,R0lGODlhDwARAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgALAAsAhAAAAAAAAB0AAB0AHQAAMx0dSB0zWgAzWh1GbDMAADIAMjNbgEgcAEceM1ozAF1/f0huf2xGHX9uSH9/XWaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwUwICBKAgIEwSKKhFgAUQtUcrQOImMCzSrjIsNqB7SxVonVUSlABXZKgCMaVVCVh2sIADs=)
(x); G
![](data:image/gif;base64,R0lGODlhCwAXAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMADAAFAAgAhAAAAAAAAAAzWh1GbDMAADIAMjNbgFozHVtISF1/f0huf2xGHX9ZM25GRn9/XWaIiIBbMwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwUSICCORDAAjViMwMG+xiiMQQ2EADs=)
(x) = f(x) -
![](data:image/gif;base64,R0lGODlhDwARAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgALAAsAhAAAAAAAAB0AAB0AHQAAMx0dSB0zWgAzWh1GbDMAADIAMjNbgEgcAEceM1ozAF1/f0huf2xGHX9uSH9/XWaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwUwICBKAgIEwSKKhFgAUQtUcrQOImMCzSrjIsNqB7SxVonVUSlABXZKgCMaVVCVh2sIADs=)
(x); G
![](data:image/gif;base64,R0lGODlhCQAYAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIADAAFAAgAhAAAAAAAAB0AAAAzWh1GbDMeRzNbgEceM1szAFtISEhZf11/f39uSH9/XXtqRGaIiIBuboiIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwUTICCOQkCMQIEqIhKgwAGLQWAAIQA7)
(x) = f(x)
![](data:image/gif;base64,R0lGODlhDwARAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgALAAsAhAAAAAAAAB0AAB0AHQAAMx0dSB0zWgAzWh1GbDMAADIAMjNbgEgcAEceM1ozAF1/f0huf2xGHX9uSH9/XWaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwUwICBKAgIEwSKKhFgAUQtUcrQOImMCzSrjIsNqB7SxVonVUSlABXZKgCMaVVCVh2sIADs=)
(x);
G
![](data:image/gif;base64,R0lGODlhCwAXAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIADAAGAAgAgwAAAAAAAB0zWgAzWgAzWx0zWzMAADNZf0gdAFszM1tuSHtqRH9/XX9uSIBbM4iIZgQSEMg5FwVmUHWBkEggEtQ3NVIEADs=)
(x) =
![](data:image/gif;base64,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)
(
![](data:image/gif;base64,R0lGODlhDwAPAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMAAwAJAAkAhAAAAAAAAB0dSAAzWx1GbDMAMzNbgEgdHVozAEhuf11/f2xGHXtqRH9/XWZ3d4BbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwUXICCOZHmU4jIuQesGQ2mggFC+LhnTaAgAOw==)
0).
Докажем, что G
![](data:image/gif;base64,R0lGODlhCAAXAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIADQAFAAcAgwAAAAAAAAAAMwAcSEgcAEceM1ozAF1/f2aIiIiIZgECAwECAwECAwECAwECAwECAwQOEBABKig2V6wtB0YwABEAOw==)
(x), G
![](data:image/gif;base64,R0lGODlhCwAXAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMADAAFAAgAhAAAAAAAAAAzWh1GbDMAADIAMjNbgFozHVtISF1/f0huf2xGHX9ZM25GRn9/XWaIiIBbMwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwUSICCORDAAjViMwMG+xiiMQQ2EADs=)
(x), G
![](data:image/gif;base64,R0lGODlhCQAYAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIADAAFAAgAhAAAAAAAAB0AAAAzWh1GbDMeRzNbgEceM1szAFtISEhZf11/f39uSH9/XXtqRGaIiIBuboiIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwUTICCOQkCMQIEqIhKgwAGLQWAAIQA7)
(x), G
![](data:image/gif;base64,R0lGODlhCwAXAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIADAAGAAgAgwAAAAAAAB0zWgAzWgAzWx0zWzMAADNZf0gdAFszM1tuSHtqRH9/XX9uSIBbM4iIZgQSEMg5FwVmUHWBkEggEtQ3NVIEADs=)
(x) – чётные функции.
Доказательство
Учитывая, что f(x) и
![](data:image/gif;base64,R0lGODlhDwARAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgALAAsAhAAAAAAAAB0AAB0AHQAAMx0dSB0zWgAzWh1GbDMAADIAMjNbgEgcAEceM1ozAF1/f0huf2xGHX9uSH9/XWaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwUwICBKAgIEwSKKhFgAUQtUcrQOImMCzSrjIsNqB7SxVonVUSlABXZKgCMaVVCVh2sIADs=)
(x) – чётные функции будем иметь:
G
![](data:image/gif;base64,R0lGODlhCAAXAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIADQAFAAcAgwAAAAAAAAAAMwAcSEgcAEceM1ozAF1/f2aIiIiIZgECAwECAwECAwECAwECAwECAwQOEBABKig2V6wtB0YwABEAOw==)
(-x) = f(-x) +
![](data:image/gif;base64,R0lGODlhDwARAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgALAAsAhAAAAAAAAB0AAB0AHQAAMx0dSB0zWgAzWh1GbDMAADIAMjNbgEgcAEceM1ozAF1/f0huf2xGHX9uSH9/XWaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwUwICBKAgIEwSKKhFgAUQtUcrQOImMCzSrjIsNqB7SxVonVUSlABXZKgCMaVVCVh2sIADs=)
(-x) = f(x) +
![](data:image/gif;base64,R0lGODlhDwARAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgALAAsAhAAAAAAAAB0AAB0AHQAAMx0dSB0zWgAzWh1GbDMAADIAMjNbgEgcAEceM1ozAF1/f0huf2xGHX9uSH9/XWaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwUwICBKAgIEwSKKhFgAUQtUcrQOImMCzSrjIsNqB7SxVonVUSlABXZKgCMaVVCVh2sIADs=)
(x) = G
![](data:image/gif;base64,R0lGODlhCAAXAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIADQAFAAcAgwAAAAAAAAAAMwAcSEgcAEceM1ozAF1/f2aIiIiIZgECAwECAwECAwECAwECAwECAwQOEBABKig2V6wtB0YwABEAOw==)
(x);
G
![](data:image/gif;base64,R0lGODlhCwAXAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMADAAFAAgAhAAAAAAAAAAzWh1GbDMAADIAMjNbgFozHVtISF1/f0huf2xGHX9ZM25GRn9/XWaIiIBbMwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwUSICCORDAAjViMwMG+xiiMQQ2EADs=)
(-x) = f(-x) -
![](data:image/gif;base64,R0lGODlhDwARAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgALAAsAhAAAAAAAAB0AAB0AHQAAMx0dSB0zWgAzWh1GbDMAADIAMjNbgEgcAEceM1ozAF1/f0huf2xGHX9uSH9/XWaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwUwICBKAgIEwSKKhFgAUQtUcrQOImMCzSrjIsNqB7SxVonVUSlABXZKgCMaVVCVh2sIADs=)
(-x) = f(x) -
![](data:image/gif;base64,R0lGODlhDwARAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgALAAsAhAAAAAAAAB0AAB0AHQAAMx0dSB0zWgAzWh1GbDMAADIAMjNbgEgcAEceM1ozAF1/f0huf2xGHX9uSH9/XWaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwUwICBKAgIEwSKKhFgAUQtUcrQOImMCzSrjIsNqB7SxVonVUSlABXZKgCMaVVCVh2sIADs=)
(x) = G
![](data:image/gif;base64,R0lGODlhCwAXAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMADAAFAAgAhAAAAAAAAAAzWh1GbDMAADIAMjNbgFozHVtISF1/f0huf2xGHX9ZM25GRn9/XWaIiIBbMwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwUSICCORDAAjViMwMG+xiiMQQ2EADs=)
(x);
G
![](data:image/gif;base64,R0lGODlhCQAYAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIADAAFAAgAhAAAAAAAAB0AAAAzWh1GbDMeRzNbgEceM1szAFtISEhZf11/f39uSH9/XXtqRGaIiIBuboiIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwUTICCOQkCMQIEqIhKgwAGLQWAAIQA7)
(-x) = f(-x)
![](data:image/gif;base64,R0lGODlhDwARAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgALAAsAhAAAAAAAAB0AAB0AHQAAMx0dSB0zWgAzWh1GbDMAADIAMjNbgEgcAEceM1ozAF1/f0huf2xGHX9uSH9/XWaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwUwICBKAgIEwSKKhFgAUQtUcrQOImMCzSrjIsNqB7SxVonVUSlABXZKgCMaVVCVh2sIADs=)
(-x) = f(x)
![](data:image/gif;base64,R0lGODlhDwARAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAgALAAsAhAAAAAAAAB0AAB0AHQAAMx0dSB0zWgAzWh1GbDMAADIAMjNbgEgcAEceM1ozAF1/f0huf2xGHX9uSH9/XWaIiIBbM4iIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwUwICBKAgIEwSKKhFgAUQtUcrQOImMCzSrjIsNqB7SxVonVUSlABXZKgCMaVVCVh2sIADs=)
(x) = G
![](data:image/gif;base64,R0lGODlhCQAYAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIADAAFAAgAhAAAAAAAAB0AAAAzWh1GbDMeRzNbgEceM1szAFtISEhZf11/f39uSH9/XXtqRGaIiIBuboiIZgECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwUTICCOQkCMQIEqIhKgwAGLQWAAIQA7)
(x);
G
![](data:image/gif;base64,R0lGODlhCwAXAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIADAAGAAgAgwAAAAAAAB0zWgAzWgAzWx0zWzMAADNZf0gdAFszM1tuSHtqRH9/XX9uSIBbM4iIZgQSEMg5FwVmUHWBkEggEtQ3NVIEADs=)
(-x) =
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=
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= G
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(x).
Свойство 4. Если y = f(x) – чётная функция, а y =
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(x) – нечётная функция, то их произведение является нечётной функцией.
Пусть функции y = f(x) и y =
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(x) имеют область определения X, симметричную относительно нуля, причём по определению
F(-x) = f(x),
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(-x) = -
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(x).
Обозначим произведение данных функций Q(x) = f(x)
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(x). Докажем, что Q(x) функция нечётная.
Доказательство
Учитывая, что f(x) – функция чётная, а
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(x) – функция нечётная, будем иметь:
Q(-x) = f(-x)
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(-x) = f(x) (-
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(x)) = -f(x)
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(x) = -Q(x).
Полученное равенство означает, что функция Q(x) нечётная.
Свойство 5. Всякую функцию, определённую на множестве X, симметричную относительно нуля, можно представить в виде суммы двух функций, каждая из которых определена на том же множестве X и одна из которых чётная, а другая нечётная.
Доказательство
Пусть функция y = f(x) имеет область определения X, симметричную относительно нуля.
Покажем, что существуют функции y =
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(x) и y =
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(x), каждая из которых определена на том же множестве X, и они такие, что
y =
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(x) +
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(x) = f(x), где y =
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(x) – чётная функция, а y =
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(x) – нечётная функции.
Положим
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(x) =
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;
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(x) =
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.