(本小題滿分12分)已知函數(shù)
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,且函數(shù)
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與
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的圖像關(guān)于直線
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對稱,又
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,
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.
(Ⅰ) 求
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的值域;
(Ⅱ) 是否存在實數(shù)m,使得命題
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和
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滿足復合命題
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為真命題?若存在,求出m的取值范圍;若不存在,說明理由.
(Ⅰ)
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的值域為
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(Ⅱ)存在實數(shù)m,使得命題
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為真命題,且m的取值范圍為
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.
解:(Ⅰ)依題意
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互為反函數(shù),由
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得
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,得
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……………………3分
故
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在
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上是減函數(shù)
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即
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的值域為
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. …………6分
(Ⅱ)由(Ⅰ)知
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是
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上的減函數(shù),
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是
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上的減函數(shù),
又
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…………9分
故
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解得
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因此,存在實數(shù)m,使得命題
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為真命題,且m的取值范圍為
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. ……………………12分
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