# Question #bbfbd

Aug 1, 2016

Domain: $x \in \mathbb{R} \text{\} \left\{1\right\}$

Range: $f \left(x\right) \in \mathbb{R} \text{\} \left\{0\right\}$

#### Explanation:

I'm assuming that your function looks like this

$f \left(x\right) = \frac{2}{x - 1}$

The domain of the function includes any value of $x$ for which $f \left(x\right)$ is defined. This implies that the domain of the function will not include values of $x$ for which the denominator is equal to zero.

$x - 1 \ne 0 \implies x \ne 1$

Therefore, you can say that the domain of the function will include any value of $x \in \mathbb{R}$ with the exception of $x = 1$, since that value would cause

$f \left(1\right) = \frac{2}{1 - 1} = \frac{2}{0} \to$ undefined

The domain will thus be $x \in \mathbb{R} \text{\} \left\{1\right\}$.

The range of the function includes any value of $f \left(x\right)$ that can be produced by plugging in the accepted values of $x$. Notice that in your case, you have no way of getting

$f \left(x\right) = 0$

That is the case because a fraction can only be equal to zero if its numerator is equal to zero.

Here the numerator of the fraction is equal to $2$ regardless of the value of $x$. The range of the function will thus be $f \left(x\right) \in \mathbb{R} \text{\} \left\{0\right\}$.

graph{2/(x-1) [-10, 10, -5, 5]}

You can use the exact same approach for the function

$f \left(x\right) = \frac{2}{x} - 1$

This time, the value of $x$ excluded from the domain will not be $x = 1$, it will be $x = 0$, since

$f \left(0\right) = \frac{2}{0} - 1 \to$ undefined

The range of the function will include $f \left(x\right) = 0$, but it will no longer include $f \left(x\right) = - 1$

graph{2/x - 1 [-10, 10, -5, 5]}