### Video Transcript

Simplify the function ๐ of ๐ฅ equals ๐ฅ squared plus two ๐ฅ over ๐ฅ squared minus four and find its domain.

๐ of ๐ฅ is a rational function and so the expression on the right-hand side is an algebraic fraction. To simplify this fraction, we need to look for common factors of the numerator and denominator which we can then cancel out. So our first task is to factor both the numerator and denominator. Starting with the numerator, we can see that the two terms, ๐ฅ squared and two ๐ฅ, have a common factor of ๐ฅ. ๐ฅ squared is ๐ฅ times ๐ฅ and two ๐ฅ is ๐ฅ times two. And so together they are ๐ฅ times ๐ฅ plus two where here we have applied the distributive property.

Now we move on to the denominator which is ๐ฅ squared minus four, and we notice that that is a difference of two squares. It is ๐ฅ minus two times ๐ฅ plus two. Now that the numerator and denominator are fully factored, we can see that they have a common factor of ๐ฅ plus two. We can cancel this out. And we see that the simplified form of ๐ of ๐ฅ is ๐ฅ over ๐ฅ minus two and that we canโt simplify any further.

So we have simplified the function, but now we need to find its domain. The domain of a rational function is the set of values for which its denominator is nonzero. In other words, it is the set of real numbers minus the set of values for which the denominator of a rational function is zero. If you look at the simplified function, you might be tempted to think that the only value of ๐ฅ for which the denominator is zero is ๐ฅ equals two. However, the denominator of the original function, as it was defined, is ๐ฅ squared minus four and not ๐ฅ minus two. And if you look at the factorized form of this denominator, itโs easy to see there are actually two values of ๐ฅ for which this denominator is zero, two and negative two. The domain is therefore the set of real numbers minus the set containing negative two and two.

So this is our answer: For every value of ๐ฅ in the domain of a function, ๐ of ๐ฅ is equal to ๐ฅ over ๐ฅ minus two. However, the domain of the function is the real numbers minus negative two and two. Had the function originally been defined as just ๐ฅ over ๐ฅ minus two, the domain wouldโve been bigger. It wouldโve been the real numbers minus just the set of two. If the function had originally been defined as just ๐ฅ over ๐ฅ minus two, then certainly the domain would be just the real numbers minus the set of two. Had we not excluded this negative two from the domain, then we wouldnโt be allowed to have cancelled ๐ฅ plus two on the numerator and the denominator because in effect we wouldโve been dividing by zero on both the top and bottom.

So while itโs possible often to simplify a rational function, the simplification process doesnโt change the domain of the function. And so when talking about the domain of the function, you should look at the original statement, the original definition of the function and not the simplified version that you get after simplification.