If the functions are defined as f(x) = √x and g(x) = √(1 - x), then what is the common domain of the following functions :
f + g, f - g, f/g, g/f, g - f where (f ± g)(x) = f(x) ± g(x),(f/g)x = f(x)/g(x)

Solution

Maths sir · 3 steps

Find each domain on its own, then keep only the values of x that work for all five functions.

1. Individual domains. f(x) = √x needs x ≥ 0, and g(x) = √(1 - x) needs x ≤ 1. Both are defined together only on [0, 1].
2. Quotients add a condition. Sums and differences need nothing extra. A quotient also needs a denominator that is not zero, and g(1) = 0 while f(0) = 0.

   Function	Extra condition	Domain
   f + g, f - g, g - f	none	[0, 1]
   f/g	g(x) ≠ 0, so x ≠ 1	[0, 1)
   g/f	f(x) ≠ 0, so x ≠ 0	(0, 1]

3. So. The common domain is [0, 1] ∩ [0, 1) ∩ (0, 1] = (0, 1), that is, 0 < x < 1. The answer is (c).

The idea to keep. For combined functions, start with the overlap of the individual domains. Then remove every point where a denominator becomes zero. A common domain is the intersection of all of them.

Answer from the NTA final answer key. Solution by PrepWiser.