Determine the domain and range of the function 𝑓 of 𝑥 equals one over 𝑥 minus two.
We have this function 𝑓 of 𝑥 equals one over 𝑥 minus two, but we don’t have a graph. Is there a way we can solve this algebraically? Can we solve this without graphing? Actually, we can. Something that we know about division is that we can’t divide something by zero. And that means that any time what’s in the denominator would be equal to zero. It’s not a possibility for this function.
If we set 𝑥 minus two equal to zero and solve, we get 𝑥 equal to two. Our function would not have a solution at 𝑥 equal to two. This is the only place for our 𝑥-values that this equation would not work. Because of that, we can say that our domain is all real numbers with the exception or minus two.
Something else interesting happens when we deal with these kinds of inverse functions. Our 𝑓 of 𝑥 cannot be equal to zero. No matter what we put in the denominator, we’ll never divide one by something and equal zero. It’s impossible. So we say that our range is all real numbers with the exception of zero — all reals minus zero.