If the functions are defined as f ( x ) = x and g ( x ) = 1 - x ,...

If the functions are defined as

f ( x ) = \sqrt{x} and

g ( x ) = \sqrt{1 - x},

then what is the common domain of the following functions:

f + g, f - g, f / g, g / f, g - f

, where \left(f \pm g\right)\left(x\right) = f\left(x\right) \pm g\left(x\right), \left(f / g\right)\left(x\right) = \dfrac{f ( x )}{g ( x )}

Options

A $0 \le x \le 1$
B $0 \le x < 1$
C $0 < x < 1$
D $0 < x \le 1$

Correct answer

C. $0 < x < 1$

Step-by-step solution

$f ( x ) + g ( x ) = \sqrt{x} + \sqrt{1 - x},$ domain \left[0, 1\right] $$ f ( x ) - g ( x ) = \sqrt{x} - \sqrt{1 - x}, $domain \left[0, 1\right]$

$g ( x ) - f ( x ) = \sqrt{1 - x} - \sqrt{x},$ domain \left[0, 1\right] $$ \dfrac{f ( x )}{g ( x )} = \dfrac{\sqrt{x}}{\sqrt{1 - x}}, $domain$ [ 0, 1 ) $$ \dfrac{g ( x )}{f ( x )} = \dfrac{\sqrt{1 - x}}{\sqrt{x}}, $domain$ ( 0, 1 ] $So, common domain is \left(0, 1\right)$

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