Question

Find the domain and the range of a function \[f(x)=\dfrac{\left| x-1 \right|}{x-1}\]

Answer 1

Hint: The domain of the functions is the set of all values that x can take, and the range is the set of all values that \[f(x)=\dfrac{\left| x-1 \right|}{x-1}\] can take. Denominator of a fraction cannot be zero. The modulus function \[\left| x \right|\] take the value \[x\] and \[-x\] when \[x>0\] and \[x<0\] respectively.

Complete step-by-step solution:
The given function is \[f(x)=\dfrac{\left| x-1 \right|}{x-1}\]
Consider \[f\] be the function defined from the set A to the set B.
Then A is called the domain of the function and contains all possible values x can take.
Also B is called the co-domain of the set.
Then the set of all images of the function, which will be a subset of the co-domain, is called the range of the function.
Now, consider the function given.
 \[f(x)=\dfrac{\left| x-1 \right|}{x-1}\]
Let's see what all the possible values for x are in order to find the domain.
Division by zero is not specified, as we all know.
As a result, a function's denominator cannot be zero.
 \[x-1\ne 0\]
Adding 1 on both sides, we get:
 \[x\ne 1\]
As a result, the only value that x cannot take is 3.
The domain of this expression is the set of all real numbers except one, which is
 \[\mathbb{R}-\left\{ 3 \right\}\]
Now, the range is the set of all the values taken by \[f(x)\]
We have \[f(x)=\dfrac{\left| x-1 \right|}{x-1}\]
Consider \[\left| x-1 \right|\]
We know that \[f(x)=\left| x \right|\] take the value \[x\] and \[-x\] when \[x>0\] and \[x<0\] respectively.
So, we have,
 \[\left| x-1 \right|=x-1\] If \[x-1>0\] and \[\left| x-1 \right|=-(x-1)\] if \[x-1<0\]
Further simplifying this we get:
 \[\left| x-1 \right|=x-1\] If \[x>1\] and \[\left| x-1 \right|=-(x-1)\] if \[x<1\]
Case 1:
If \[\left| x-1 \right|=x-1\] , then \[\dfrac{\left| x-1 \right|}{x-1}=\dfrac{x-1}{x-1}=1\]
Case 2:
If \[\left| x-1 \right|=-(x-1)\] then, \[\dfrac{\left| x-1 \right|}{x-1}=\dfrac{-\left( x-1 \right)}{x-1}=-1\]
That is,
 \[f(x)=1\] If \[x>1\] and \[f(x)=-1\] If \[x<1\]
So \[f(x)\] takes two values \[1\] and \[-1\]
This gives the range of the function as \[\left\{ -1,1 \right\}\] and the domain is R-{1}.

Note: The codomain and domain of a function are also indicated when it is defined. As in this situation, the domain and co-domain do not need to be different. They could also be the same.