Question

已知定义在R上的偶函数f(x)满足f(x+1)=-f(x),且当x∈[0,1]时f(x)=3^x-1,求f(log(3)

Answer 1

log_3(9) < log_3(26) < log_3(27)

2 < log_3(26) < 3

0 < log_3(26) - 2 < 1

f(x + 1) = - f(x),令x = x + 1

f(x + 2) = - f(x + 1)

f(x + 2) = - (- f(x)) = f(x),令x = x - 2

f(x - 2) = f(x)

∴f(log_3(26)) = f(log_3(26) - 2)

= 3^[log_3(26) - 2] - 1

= 3^log_3(26) • 3^(- 2) - 1

= 26 • 1/9 - 1

= 17/9