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

Given:

Need to find: Where the functions are defined.

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

Therefore,

x – 3 = 0

⇒ x = 3

It means that the denominator is zero when x = 3

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

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

Now if we put any value of x from the domain set the output value will be either (-ve) or (+ve), but the value will never be 6

So, the range of the function is the set of all the real numbers except 6.

The range of the function, R_{f(x)} = (-∞, 6) ∪(6, ∞).

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