If the functions are defined as [f(x) = x ] and [g(x) = 1 - x ], then what is the common d…

Question

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 $$(f \pm g)(x) = f(x) \pm g(x),(f/g)x = {{f(x)} \over {g(x)}}$$

Accepted Answer

$$f + g = \sqrt x  + \sqrt {1 - x} $$

$$ \Rightarrow x \ge 0$$ \& $$1 - x \ge 0 \Rightarrow x \in [0,1]$$

$$f - g = \sqrt x  - \sqrt {1 - x} $$

$$ \Rightarrow x \ge 0$$ \& $$1 - x \ge 0 \Rightarrow x \in [0,1]$$

$$f/g = {{\sqrt x } \over {\sqrt {1 - x} }}$$

$$ \Rightarrow x \ge 0$$ \& $$1 - x > 0 \Rightarrow x \in [0,1)$$

$$g/f = {{\sqrt {1 - x} } \over {\sqrt x }}$$

$$ \Rightarrow 1 - x \ge 0$$ \& $$x > 0 \Rightarrow x \in (0,1]$$

$$g - f = \sqrt {1 - x}  - \sqrt x $$

$$ \Rightarrow 1 - x \ge 0$$ \& $$x \ge 0 \Rightarrow x \in [0,1]$$

$$ \Rightarrow $$ $$x \in (0,1)$$

Solutions · 1