What is the domain of the function
𝑦 equals 𝑥 squared minus one over 𝑥 squared plus one?
Remember, the domain of a function
is the set of all possible inputs to that function. And if we inspect our function 𝑦
equals 𝑥 squared minus one over 𝑥 squared plus one carefully, we can see it’s a
rational function. That is, it’s the quotient of a
pair of polynomials. So we’ll remind ourselves what we
know about the domain of a rational function. The domain of a rational function
is the set of real numbers, but we exclude any values of 𝑥 that make the
denominator of the function equal to zero. In this case then, the function is
undefined for any values of 𝑥 which satisfy the equation 𝑥 squared plus one equals
zero. To establish which values of 𝑥
this is true for, let’s solve this equation.
We can begin by subtracting one
from both sides, so 𝑥 squared is equal to negative one. Then, we would look to take both
the positive and negative square root of negative one. But of course the square root of a
negative number is not a real number. And since we said the domain of a
rational function is just the set of real numbers, there are no values of 𝑥 in this
case which are going to make the denominator equal to zero. We can therefore say that the
domain of our function and the set of numbers that ensure it is well defined is
simply the set of real numbers.