Answer
(a) $\sqrt{x + 7} + \sqrt{x - 2}$; Domain = $[x | x\geq 2,$ $x$ $belongs$ $to$ $the$ $Real$ $numbers]$
(b) $\sqrt{x + 7} - \sqrt{x - 2}$; Domain = $[x | x\geq 2,$ $x$ $belongs$ $to$ $the$ $Real$ $numbers]$
(c) $\sqrt{x^2 + 5x - 14}$; Domain = $[x | (x\leq -7)$U$(x\geq 2),$ $x$ $belongs$ $to$ $the$ $Real$ $numbers]$
(d) $\frac{\sqrt{x^2 + 5x - 14}}{x-2}$; Domain = $[x | (x\leq -7)$U$(x\gt 2),$ $x$ $belongs$ $to$ $the$ $Real$ $numbers]$
Work Step by Step
$$f(x) = \sqrt{x + 7}$$ $$g(x) = \sqrt{x - 2}$$
(a) $(f + g)(x)$ $$\sqrt{x + 7} + \sqrt{x - 2}$$ which cannot be further simplified. Since this is a radical function, to find the domain we must find the values of $x$ where the radicals are greater than or equal to zero: $$x + 7 \geq 0$$ $$AND$$ $$x - 2 \geq 0$$ Solving each gives: $$x \geq -7$$ $$AND$$ $$x\geq 2$$ Since the only values of $x$ that truly satisfy the conditions are $x\geq 2$, the $Domain$ is $all$ $Real$ $numbers$ $greater$ $than$ $or$ $equal$ $to$ $2$, or also, $[x | x\geq 2,$ $x$ $belongs$ $to$ $the$ $Real$ $numbers]$
(b) $(f - g)(x)$ $$\sqrt{x + 7} - \sqrt{x - 2}$$ which cannot be further simplified. The domain is found in the same manner as above. Therefore, the $Domain$ is $[x | x\geq 2,$ $x$ $belongs$ $to$ $the$ $Real$ $numbers]$
(c) $(fg)(x)$ $$\sqrt{x + 7} \times \sqrt{x -2}$$ $$\sqrt{(x + 7)(x-2)}$$ $$\sqrt{x^2 + 5x - 14}$$ To find the domain, we use the factors of the polynomial inside the radical function and we find the intervals of $x$ where both factors, when multiplied yield a positive value. This condition is satisfied only when $x\leq -7$ and $x\geq 2$. Therefore, the $Domain$ is $[x | (x\leq -7)$U$(x\geq 2),$ $x$ $belongs$ $to$ $the$ $Real$ $numbers]$
(d) $\frac{f}{g}(x)$ $$\frac{\sqrt{x + 7}}{\sqrt{x-2}}$$ $$\frac{\sqrt{(x+7)(x-2)}}{x-2}$$ $$\frac{\sqrt{x^2 + 5x - 14}}{x-2}$$ To find the domain, we have two limitations: a denominator that must be greater than zero, and a radical whose polynomial's factors, when multiplied, must be equal to or greater than zero. For the denominator, we can say $$x-2 \neq 0$$ $$x \neq 2$$. For the radical, we can apply the same logic we used in part (c) and find that $x\leq -7$ and $x\geq 2$. Combining both, we find that the $Domain$ is $[x | (x\leq -7)$U$(x\gt 2),$ $x$ $belongs$ $to$ $the$ $Real$ $numbers]$
