The domain is the set of all possible
x-values which will make the function valid.
![f(x) = \frac{3}{x-2} \ \ \ \ , \ g(x) = \sqrt{x-1}](https://tex.z-dn.net/?f=f%28x%29%20%3D%20%5Cfrac%7B3%7D%7Bx-2%7D%20%5C%20%5C%20%5C%20%5C%20%2C%20%5C%20g%28x%29%20%3D%20%20%5Csqrt%7Bx-1%7D%20)
For the given function :
The denominator of a fraction cannot be
zero
The number under a square root sign must be
Non negative
(a)(1) The domain of f ⇒⇒⇒ R - {2}
Because ⇒⇒⇒ x - 2 = 0 ⇒⇒⇒ x = 2
(2) The domain of g ⇒⇒⇒ [1,∞)
Because: x - 1 ≥ 0 ⇒⇒⇒ x ≥ 1
(3)
![f + g = \frac{3}{x-2} + \sqrt{x-1}](https://tex.z-dn.net/?f=f%20%2B%20g%20%3D%20%5Cfrac%7B3%7D%7Bx-2%7D%20%2B%20%5Csqrt%7Bx-1%7D%20)
The domain of (f+g) ⇒⇒⇒ [1,∞) - {2}
because: x-2 = 0 ⇒⇒⇒ x = 2 and x - 1 ≥ 0 ⇒⇒⇒ x ≥ 1
(4)
![f - g = \frac{3}{x-2} - \sqrt{x-1}](https://tex.z-dn.net/?f=f%20-%20g%20%3D%20%5Cfrac%7B3%7D%7Bx-2%7D%20-%20%5Csqrt%7Bx-1%7D)
The domain of (f-g) ⇒⇒⇒ [1,∞) - {2}
because: x-2 = 0 ⇒⇒⇒ x = 2 and x - 1 ≥ 0 ⇒⇒⇒ x ≥ 1
(5)
![f * g = \frac{3}{x-2} * \sqrt{x-1} = \frac{3 \sqrt{x-1}}{(x-2)}](https://tex.z-dn.net/?f=f%20%2A%20g%20%3D%20%5Cfrac%7B3%7D%7Bx-2%7D%20%2A%20%5Csqrt%7Bx-1%7D%20%3D%20%5Cfrac%7B3%20%5Csqrt%7Bx-1%7D%7D%7B%28x-2%29%7D%20)
The domain of (f*g) ⇒⇒⇒ [1,∞) - {2}
because: x-2 = 0 ⇒⇒⇒ x = 2 and x - 1 ≥ 0 ⇒⇒⇒ x ≥ 1
(6)
![f * f = \frac{3}{x-2} * \frac{3}{x-2} = \frac{9}{(x-2)^2}](https://tex.z-dn.net/?f=f%20%2A%20f%20%3D%20%5Cfrac%7B3%7D%7Bx-2%7D%20%2A%20%5Cfrac%7B3%7D%7Bx-2%7D%20%3D%20%5Cfrac%7B9%7D%7B%28x-2%29%5E2%7D)
The domain of ff ⇒⇒⇒ R - {2}
Because ⇒⇒⇒ x-2 = 0 ⇒⇒⇒ x = 2
(7)
![\frac{f}{g} = \frac{\frac{3}{x-2} }{ \sqrt{x-1} } = \frac{3}{(x-2) \sqrt{x-1}}](https://tex.z-dn.net/?f=%20%5Cfrac%7Bf%7D%7Bg%7D%20%3D%20%20%20%5Cfrac%7B%5Cfrac%7B3%7D%7Bx-2%7D%20%7D%7B%20%5Csqrt%7Bx-1%7D%20%7D%20%3D%20%20%5Cfrac%7B3%7D%7B%28x-2%29%20%5Csqrt%7Bx-1%7D%7D%20)
The domain of (f/g) ⇒⇒⇒ (1,∞) - {2}
because: x-2 = 0 ⇒⇒⇒ x = 2 and x - 1 > 0 ⇒⇒⇒ x > 1
(8)
![\frac{g}{f} = \frac{ \sqrt{x-1} }{ \frac{3}{x-2} } = \frac{1}{3} (x-2) \sqrt{x-1}](https://tex.z-dn.net/?f=%20%5Cfrac%7Bg%7D%7Bf%7D%20%3D%20%20%5Cfrac%7B%20%5Csqrt%7Bx-1%7D%20%7D%7B%20%5Cfrac%7B3%7D%7Bx-2%7D%20%7D%20%3D%20%20%5Cfrac%7B1%7D%7B3%7D%20%28x-2%29%20%5Csqrt%7Bx-1%7D%20)
The domain of (g/f) ⇒⇒⇒ [1,∞) - {2}
Because: x - 2 = 0 ⇒⇒⇒ x = 2 and x - 1 ≥ 0 ⇒⇒⇒ x ≥ 1
===================================================
(b)
(9)
![(f + g)(x) = \frac{3}{x-2} + \sqrt{x-1}](https://tex.z-dn.net/?f=%20%28f%20%2B%20g%29%28x%29%20%3D%20%5Cfrac%7B3%7D%7Bx-2%7D%20%2B%20%5Csqrt%7Bx-1%7D%20)
(10)
![(f - g)(x) = \frac{3}{x-2} - \sqrt{x-1}](https://tex.z-dn.net/?f=%28f%20-%20g%29%28x%29%20%3D%20%5Cfrac%7B3%7D%7Bx-2%7D%20-%20%5Csqrt%7Bx-1%7D)
(11)
![(f * g)(x) = \frac{3}{x-2} * \sqrt{x-1} = \frac{3 \sqrt{x-1}}{(x-2)}](https://tex.z-dn.net/?f=%28f%20%2A%20g%29%28x%29%20%3D%20%5Cfrac%7B3%7D%7Bx-2%7D%20%2A%20%5Csqrt%7Bx-1%7D%20%3D%20%5Cfrac%7B3%20%5Csqrt%7Bx-1%7D%7D%7B%28x-2%29%7D%20)
(12)
![(f * f)(x) = \frac{3}{x-2} * \frac{3}{x-2} = \frac{9}{(x-2)^2}](https://tex.z-dn.net/?f=%28f%20%2A%20f%29%28x%29%20%3D%20%5Cfrac%7B3%7D%7Bx-2%7D%20%2A%20%5Cfrac%7B3%7D%7Bx-2%7D%20%3D%20%5Cfrac%7B9%7D%7B%28x-2%29%5E2%7D)
(13)
![\frac{f}{g} = \frac{\frac{3}{x-2} }{ \sqrt{x-1} } = \frac{3}{(x-2) \sqrt{x-1}}](https://tex.z-dn.net/?f=%20%5Cfrac%7Bf%7D%7Bg%7D%20%3D%20%20%20%5Cfrac%7B%5Cfrac%7B3%7D%7Bx-2%7D%20%7D%7B%20%5Csqrt%7Bx-1%7D%20%7D%20%3D%20%20%5Cfrac%7B3%7D%7B%28x-2%29%20%5Csqrt%7Bx-1%7D%7D%20)
(14)