Find the domain and the range of each of the following real function:

1.	Find the domain and the range of each of the following real function: f(x) = \(\sqrt{\frac{x-5}{3-x}}\)
Answer» Given: f(x) = \(\sqrt{\frac{x-5}{3-x}}\) Need to find: Where the functions are defined. The condition for the function to be defined, 3 - x > 0 ⇒ x < 3 So, the domain of the function is the set of all the real numbers lesser than 3. The domain of the function, D_f(x) = (-∞, 3). The condition for the range of the function to be defined, x - 5 ≥ 0 _& 3 - x > 0 ⇒ x ≥ 5 _& x < 3 Both the conditions can’t be satisfied simultaneously. That means there is no range for the function f(x).

Find the domain and the range of each of the following real function: f(x) = \(\sqrt{\frac{x-5}{3-x}}\)

Answer»

Given: f(x) = \(\sqrt{\frac{x-5}{3-x}}\)

Need to find: Where the functions are defined.

The condition for the function to be defined,

3 - x > 0

⇒ x < 3

So, the domain of the function is the set of all the real numbers lesser than 3.

The domain of the function, D_f(x) = (-∞, 3).

The condition for the range of the function to be defined,

x - 5 ≥ 0 _& 3 - x > 0

⇒ x ≥ 5 _& x < 3

Both the conditions can’t be satisfied simultaneously. That means there is no range for the function f(x).