Video: Simplifying and Determining the Domain of a Sum of Two Rational Functions

Simplify the function 𝑛(π‘₯) = (2/(π‘₯ βˆ’ 8)) + (4/(8 βˆ’ π‘₯)), and determine its domain.

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Video Transcript

Simplify the function 𝑛 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 doesn’t change. And we’re 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 𝑛 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 can’t 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.

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