Olimpíadas ⇒ Funções Tópico resolvido

[PNL123]

Seja [tex3]M(x)=\frac{1}{2}(a^{x}+a^{-x})~~e ~~N(x)=\frac{1}{2}(a^{x}-a^{-x})[/tex3]

Código: Selecionar tudoSHIFT+Click na eq = ZOOM

[tex3]M(x)=\frac{1}{2}(a^{x}+a^{-x})~~e ~~N(x)=\frac{1}{2}(a^{x}-a^{-x})[/tex3]

.Demontre que:
[tex3]M(x+y)=M(x)M(y)+N(x)N(y)~~~~e~~~~N(x+y)=M(x)N(y)+M(y)N(x)[/tex3]

Código: Selecionar tudoSHIFT+Click na eq = ZOOM

[tex3]M(x+y)=M(x)M(y)+N(x)N(y)~~~~e~~~~N(x+y)=M(x)N(y)+M(y)N(x)[/tex3]

Não possuo Gabarito

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[petras]

[tex3]\frac{1}{2}(a^{x+y}+a^{-x-y})=[\frac{1}{2}(a^{x}+a^{-x})][\frac{1}{2}(a^{y}+a^{-y})]+[\frac{1}{2}(a^{x}-a^{-x})][\frac{1}{2}(a^{y}-a^{-y})] [/tex3]

Código: Selecionar tudoSHIFT+Click na eq = ZOOM

[tex3]\frac{1}{2}(a^{x+y}+a^{-x-y})=[\frac{1}{2}(a^{x}+a^{-x})][\frac{1}{2}(a^{y}+a^{-y})]+[\frac{1}{2}(a^{x}-a^{-x})][\frac{1}{2}(a^{y}-a^{-y})] [/tex3]

[tex3]\frac{1}{2}(a^{x+y}+a^{-x-y})=\frac{1}{4}(a^{x+y}+a^{x-y}+a^{-x+y}+a^{-x-y})+\frac{1}{4}(a^{x+y}-a^{x-y}-a^{-x+y}+a^{-x-y}) [/tex3]

Código: Selecionar tudoSHIFT+Click na eq = ZOOM

[tex3]\frac{1}{2}(a^{x+y}+a^{-x-y})=\frac{1}{4}(a^{x+y}+a^{x-y}+a^{-x+y}+a^{-x-y})+\frac{1}{4}(a^{x+y}-a^{x-y}-a^{-x+y}+a^{-x-y}) [/tex3]

[tex3]\frac{1}{2}(a^{x+y}+a^{-x-y})=\frac{1}{4}(2a^{x+y}+2a^{-x-y}) \rightarrow \frac{1}{2}(a^{x+y}+a^{-x-y})=\frac{1}{2}(a^{x+y}+a^{-x-y}) ~c.q.d.[/tex3]

Código: Selecionar tudoSHIFT+Click na eq = ZOOM

[tex3]\frac{1}{2}(a^{x+y}+a^{-x-y})=\frac{1}{4}(2a^{x+y}+2a^{-x-y}) \rightarrow \frac{1}{2}(a^{x+y}+a^{-x-y})=\frac{1}{2}(a^{x+y}+a^{-x-y}) ~c.q.d.[/tex3]

[tex3]\frac{1}{2}(a^{x+y}-a^{-x-y})=[\frac{1}{2}(a^{x}+a^{-x})][\frac{1}{2}(a^{y}-a^{-y})]+[\frac{1}{2}(a^{y}+a^{-y})][\frac{1}{2}(a^{x}-a^{-x})] [/tex3]

Código: Selecionar tudoSHIFT+Click na eq = ZOOM

[tex3]\frac{1}{2}(a^{x+y}-a^{-x-y})=[\frac{1}{2}(a^{x}+a^{-x})][\frac{1}{2}(a^{y}-a^{-y})]+[\frac{1}{2}(a^{y}+a^{-y})][\frac{1}{2}(a^{x}-a^{-x})] [/tex3]

[tex3]\frac{1}{2}(a^{x+y}+a^{-x-y})=\frac{1}{4}(a^{x+y}-a^{x-y}+a^{-x+y}-a^{-x-y})+\frac{1}{4}(a^{x+y}-a^{-x+y}+a^{x-y}-a^{-x-y}) [/tex3]

Código: Selecionar tudoSHIFT+Click na eq = ZOOM

[tex3]\frac{1}{2}(a^{x+y}+a^{-x-y})=\frac{1}{4}(a^{x+y}-a^{x-y}+a^{-x+y}-a^{-x-y})+\frac{1}{4}(a^{x+y}-a^{-x+y}+a^{x-y}-a^{-x-y}) [/tex3]

[tex3]\frac{1}{2}(a^{x+y}-a^{-x-y})=\frac{1}{4}(2a^{x+y}-2a^{-x-y}) \rightarrow \frac{1}{2}(a^{x+y}+a^{-x-y})=\frac{1}{2}(a^{x+y}-a^{-x-y}) ~c.q.d.[/tex3]

Código: Selecionar tudoSHIFT+Click na eq = ZOOM

[tex3]\frac{1}{2}(a^{x+y}-a^{-x-y})=\frac{1}{4}(2a^{x+y}-2a^{-x-y}) \rightarrow \frac{1}{2}(a^{x+y}+a^{-x-y})=\frac{1}{2}(a^{x+y}-a^{-x-y}) ~c.q.d.[/tex3]