Question

已知函數(shù)
（1）求函數(shù)的定義域和值域；（2）若函數(shù)有最小值為，求的值。

Answer 1

(1)定義域?yàn)?img src="http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031958132485.png" style="vertical-align:middle;" />,當(dāng)時(shí),值域?yàn)?img src="http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031958163694.png" style="vertical-align:middle;" />,當(dāng)時(shí),值域?yàn)?img src="http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031958194710.png" style="vertical-align:middle;" />;
(2)

試題分析：(1)根據(jù)對數(shù)函數(shù)的定義域?yàn)?img src="http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031958241566.png" style="vertical-align:middle;" />,則由函數(shù),可得,解之得,從而可得所求函數(shù)的定義域?yàn)?img src="http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031958132485.png" style="vertical-align:middle;" />;根據(jù)對數(shù)函數(shù)當(dāng)時(shí)為單調(diào)遞增函數(shù),當(dāng)時(shí)為單調(diào)遞減函數(shù),又由復(fù)合函數(shù)的“同增異減”性質(zhì)(注:兩個(gè)復(fù)合函數(shù)的單調(diào)性相同時(shí)復(fù)合函數(shù)為單調(diào)遞增,不同時(shí)復(fù)合函數(shù)為單調(diào)遞減),可將函數(shù)對其底數(shù)分為與兩情況進(jìn)行分類討論,從而求出函數(shù)的值域;(2)由(1)知當(dāng)時(shí)函數(shù)有最小值,從而有,可解得.
試題解析：(1)由已知得,解之得,故所求函數(shù)的定義域?yàn)?img src="http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031958132485.png" style="vertical-align:middle;" />.
原函數(shù)可化為,設(shè),又,所以.
當(dāng)時(shí),有;當(dāng)時(shí), .
故當(dāng)時(shí),函數(shù)的值域?yàn)?img src="http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031958163694.png" style="vertical-align:middle;" />,當(dāng)時(shí),值域?yàn)?img src="http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824031958194710.png" style="vertical-align:middle;" />.
(2)由題意及(1)知:當(dāng)時(shí),函數(shù)有最小值,即,可解得.