I am trying to find the domain and range of:
$$f(x) = \frac {x^2 – 4x + 3 }{x – 1}$$
In the book I am using, it says that the domain (x) is the set of all real numbers except 1 and that the range (y) is the set of all real numbers except -2. However, we can simplify this equation to:
$$f(x) = \frac {(x-3)(x-1)}{x – 1}$$
which is equal to
$$f(x) = x-3$$
Now, the domain and range of this simplified equation is different; they are now the set of all real numbers.
My question is: when determining the domain and range, do we simplify or don't we? Does simplifying the equation really produce different domains and ranges? Am I doing something wrong?
Best Answer
We should always state our domain when we define our function.
The original function is $f: \mathbb{R} \setminus \{1 \} \to \mathbb{R}$
$$f(x) = \frac{(x-3)(x-1)}{x-1}.$$
It is well defiend on $\mathbb{R} \setminus \{ 1\}$ but it is not defined when $x=1$.
Compared it to function $g: \mathbb{R} \to \mathbb{R}$
$$g(x)=x-3$$
$f$ and $g$ are different functions because the domain is different.
When such question is being asked, the common practice is treat $f$ as $f$ and not $g$.