Simplify the function 𝑛 of 𝑥 equals two over 𝑥 minus eight plus four over eight minus 𝑥, and determine its domain.
Starting with this function, in order to add fractions, we need a common denominator. Is there something we can multiply eight minus 𝑥 by so that it becomes 𝑥 minus eight?
Think about it like this. We currently have a positive eight and a negative 𝑥. And we want to have a positive 𝑥 and a negative eight. If we multiply eight minus 𝑥 by negative one, it becomes 𝑥 minus eight. And to change that, we would need to multiply four over eight minus 𝑥 by negative one over negative one. Two over 𝑥 minus eight doesn’t change. And we’re adding negative four over 𝑥 minus eight. Now that the fractions have a common denominator, we can subtract. Two minus four equals negative two. And the denominator stays 𝑥 minus eight.
The function 𝑛 of 𝑥 can be written as negative two over 𝑥 minus eight. The domain represents what values 𝑥 can be. And we know that this denominator 𝑥 minus eight cannot be equal to zero because we can’t divide by zero. 𝑥 minus eight cannot be equal to zero. So we add eight to both sides. And that tells us that 𝑥 cannot be eight. Eight is not part of the domain. We can say that the domain equals all real numbers except eight.