Question

已知函數(shù)y=f（x）的圖象如圖，則滿足f(log2(x-1))•f(2-x2-1)≥0的x的取值范圍為

（1，3]

．

Answer 1

分析：先利用圖象，將所求不等式等價(jià)轉(zhuǎn)化為對(duì)數(shù)不等式組或指數(shù)不等式組，再利用指數(shù)函數(shù)與對(duì)數(shù)函數(shù)的性質(zhì)解不等式組即可得所求不等式的解集

解答：解：由圖可知，x≤1時(shí)，f（x）≥0，x＞1時(shí)，f（x）＜0
   f(log2(x-1))•f(2-x2-1)≥0
?

f(log2(x-1))≥0 f(2-x2-1)≥0

或

f(log2(x-1))≤0 f(2-x2-1)≤0

?

log2(x-1)≤1 2-x2-1≤1

或

log2(x-1)≥1 2-x2-1≥1

?

0＜x-1≤2 -x2-1≤0

或

x-1≥2 -x2-1≥0

?

1＜x≤3 x∈R

或

x≥3 x∈∅

?1＜x≤3
故答案為（1，3]

點(diǎn)評(píng)：本題考查了利用函數(shù)圖象解不等式，指數(shù)函數(shù)與對(duì)數(shù)函數(shù)的性質(zhì)，簡(jiǎn)單的指數(shù)、對(duì)數(shù)不等式的解法