Find the domain and the range of each of the following real function: f(x) = |x - 4|/x - 4

Question

Find the domain and the range of each of the following real function: f(x) = \(\frac{|x-4|}{x-4}\)

Answer 1

Given: f(x) = \(\frac{|x-4|}{x-4}\)

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 }