# What is the domain and range of y= 4 / (x^2-1)?

Sep 20, 2015

Domain: $\left(- \infty , - 1\right) \cup \left(- 1 , 1\right) \cup \left(1 , \infty\right)$
Range: $\left(- \infty , - 4\right] \cup \left(0 , \infty\right)$

#### Explanation:

Best explained through the graph.
graph{4/(x^2-1) [-5, 5, -10, 10]}

We can see that for the domain, the graph starts at negative infinity. It then hits a vertical asymptote at x = -1.

That's fancy math-talk for the graph is not defined at x = -1, because at that value we have $\frac{4}{{\left(- 1\right)}^{2} - 1}$ which equals $\frac{4}{1 - 1}$ or $\frac{4}{0}$.

Since you can't divide by zero, you can't have a point at x = -1, so we keep it out of the domain (recall that the domain of a function is the collection of all the x-values that produce a y-value).

Then, between -1 and 1, everything's fine, so we have to include it in the domain.

Things start getting funky at x = 1 again. Once more, when you plug in 1 for x, the result is $\frac{4}{0}$ so we have to exclude that from the domain.

To sum it up, the function's domain is from negative infinity to -1, then from -1 to 1, and then to infinity. The mathy way of expressing that is $\left(- \infty , - 1\right) \cup \left(- 1 , 1\right) \cup \left(1 , \infty\right)$.

The range follows the same idea: it's the set of all y-values of the function. We can see from the graph that from negative infinity to -4, all is well.

Then things start going south. At y=-4, x=0; but then, if you try y=-3, you won't get an x. Watch:

$- 3 = \frac{4}{{x}^{2} - 1}$

$- 3 \left({x}^{2} - 1\right) = 4$

${x}^{2} - 1 = - \frac{4}{3}$

${x}^{2} = - \frac{4}{3} + 1 = - \frac{1}{3}$

$x = \sqrt{- \frac{1}{3}}$

There is no such thing as the square root of a negative number. That's saying some number squared equals $- \frac{1}{3}$, which is impossible because squaring a number always has a positive result.

That means $y = \text{-} 3$ is undefined and so is not part of our range. The same is true for all y-values between 4 and 0.

From 0 above, everything is good all the way to infinity. Our range is then negative infinity to -4, then 0 to infinity; in math terms, $\left(- \infty , - 4\right] \cup \left(0 , \infty\right)$.

In general, to find domain and range, you have to look for places where things are suspicious. That usually involves stuff like dividing by zero, taking the square root of a negative number, etc.

Whenever you find a point like this, remove it from the domain/range and build up your interval notation.