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 – 4 = 0

⇒ x = 4

It means that the denominator is zero when x = 4

So, the domain of the function is the set of all the real numbers except 4.

The domain of the function, D_{f(x)} = (- ∞, 4) ∪ (4, ∞).

The numerator is an absolute function of the denominator. So, for any value of x from the domain set, we always get either +1 or -1 as the output. So, the range of the function is a set containing -1 and +1

Therefore, the range of the function, R_{f(x)} = { -1 , 1 }

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