课本上有一道习题:设f(x)=e~x-e~-x/2,g(x)=求证:(1)[g(x)]~2-[f(x)]~2=1.(2)f(2x)=2f(x)g(x).(3)g(2x)=[f(x)]~2+[g(x)]~2.上述习题比较容易证明.事实上,该题的知识背景就是高等数学中定义的一种基本函数.对于中学生,我们不妨把f(x)=e~x-e~(-x)/2称为双曲正弦函数,记 There is an exercise in the textbook: Let f (x) = e ~ xe ~ -x / 2, g (x) = Prove that: (2) f (2x) = 2f (x) g (x) (3) g (2x) = [f (x)] ~ 2 + [g (x)] ~ 2 The above problem is relatively easy to prove. In fact, the knowledge background of the question is a basic function defined in advanced mathematics. For middle school students, we may call f (x) = e ~ xe ~ (-x) / 2 as the hyperbolic sine function