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,

x + 2 = 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_{1}) we need to equate the denominator of the function to 0.

Therefore,

3 – x = 0

⇒ x = 3

It means that the denominator is zero when x = 3

So, the range of the function is the set of all the real numbers except 3.

The range of the function, R_{f(x)} = (- ∞, 3) ∪ (3, ∞).

1