题目设x,y,z∈(0,+∞)且x2+y2+z2=1,求函数f=x+y+z-xyz的值域.这是一道美国数学月刊征解题,文[1]给出此题一个简解.本文给出此题一个以三角代换为突破口的简单解法,其思路更自然流畅、过程更简捷明了.证明由已知条件可设x=cosα,y=sinαcosβ,z=sinαsinβ,其中α,β∈(0,π2), The subject set x, y, z ∈ (0, + ∞) and x2 + y2 + z2 = 1, find the value of the function f = x + y + z-xyz.This is a US mathematical monthly solvable problem, the article [ 1] gives a brief explanation of this problem.This paper gives a simple solution to the problem of triangular substitution as a breakthrough, the idea of ​​a more natural and fluent, the process more concise and clear. Prove that the known conditions can be set x = cosα, y = sinαcosβ, z = sinαsinβ, where α, β∈ (0, π2),