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

Given:

Need to find: Where the functions are defined.

Let, ---- (1)

To find the domain of the function f(x) we need to equate the denominator of the function to 0.

Therefore,

2 – x = 0

⇒ x = 2

It means that the denominator is zero when x = 2

So, the domain of the function is the set of all the real numbers except 2.

The domain of the function, D_f(x) = (- ∞, 2) ∪ (2, ∞).

Now, to find the range of the function we need to interchange x and y in the equation no. (1)

So the equation becomes,

⇒

⇒

⇒

⇒

To find the range of the function f(x) we need to equate the denominator of the function to 0.

Therefore,

x + 1 = 0

⇒ x = -1

It means that the denominator is zero when x = -1

So, the range of the function is the set of all the real numbers except -1.

The range of the function, R_f(x) = (- ∞, -1) ∪ (-1, ∞).