What is the domain of the function tangent of negative 𝜋𝑥 over two whose graph is shown?
The domain will be all of the 𝑥-values that the graph represents. So essentially, we have to ask ourselves, is there anywhere on the 𝑥-axis that a number doesn’t have somewhere to actually land? So almost every value on our 𝑥-axis has a place to go, and it’s represented by that arrow.
So, for example, at zero, at 𝑥 equals zero, 𝑦 is equal to zero. And at 𝑥 is a positive small, small fraction, we end up below the 𝑥-axis for negative four 𝑦, and it decreases as we get closer and closer to one. However, at one, we actually can’t land anywhere where 𝑥 equals one; that is called an asymptote, and it’s a value that the graph will approach but never actually reach.
So 𝑥 equals one we’ll never reach, 𝑥 equals three we won’t reach, 𝑥 equals negative one, and 𝑥 equals negative three. So all of the 𝑥-values except for these will work. So our domain will be all real numbers except these specific values, so how do we classify these?
Well it’s not just the integers because it doesn’t include two or zero or negative two or negative four; it excludes all of those even integers, so this must be the odd integers. So our domain must be all real numbers except odd integers.