Question

Find the range Of f(x)=(1-(x²-6x+9)^1/2)^1/2

Answer 1

By factorising, we get
x²-6x+9 = (x-3)²
f(x) = (1-(x²-6x+9)^1/2)^1/2
= (1-((x-3)²)^1/2)^1/2
= (1-(x-3))^1/2

The term inside the square root must be greater than or equal to 0
So,
1-(x-3) >= 0
1-x+3 >= 0
4-x >= 0
4 >= x
x <= 4

ie, x should be less than or equal to 4
So the range = (-α , 4]