Question

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

Answer 1

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.