Video Transcript
Simplify the function π of π₯ equals two over π₯ minus eight plus four over eight minus π₯, 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 π₯ by so that it becomes π₯ minus eight?
Think about it like this. We currently have a positive eight and a negative π₯. And we want to have a positive π₯ and a negative eight. If we multiply eight minus π₯ by negative one, it becomes π₯ minus eight. And to change that, we would need to multiply four over eight minus π₯ by negative one over negative one. Two over π₯ minus eight doesnβt change. And weβre adding negative four over π₯ minus eight. Now that the fractions have a common denominator, we can subtract. Two minus four equals negative two. And the denominator stays π₯ minus eight.
The function π of π₯ can be written as negative two over π₯ minus eight. The domain represents what values π₯ can be. And we know that this denominator π₯ minus eight cannot be equal to zero because we canβt divide by zero. π₯ minus eight cannot be equal to zero. So we add eight to both sides. And that tells us that π₯ cannot be eight. Eight is not part of the domain. We can say that the domain equals all real numbers except eight.
