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

Rhodri Jones

Simplify the function 𝑛(𝑥) = −8/(𝑥 − 6) + (𝑥 − 6)/(𝑥² − 6𝑥) and determine its domain.

03:34

Video Transcript

Simplify the function 𝑛 of 𝑥 equals negative eight divided by 𝑥 minus six plus 𝑥 minus six divided by 𝑥 squared minus six 𝑥 and determine its domain.

At first glance, when we look at the function, it appears that all real numbers are valid inputs. However, when we look more closely, we can see that some values of 𝑥 would make the denominators equal to zero, which would give us undefined values.

So in order to work out which input values generate undefined outputs, we need to find the values of 𝑥 that give us zero denominators. In this case, this involves solving 𝑥 minus six equals zero and also 𝑥 squared minus six 𝑥 equals zero.

Adding six to both sides of the first equation gives us 𝑥 equals six. Therefore, an input of six would give us an undefined output. We can solve the second equation 𝑥 squared minus six 𝑥 equals zero by firstly factorising out an 𝑥. This gives us 𝑥 multiplied by 𝑥 minus six equal zero. Solving this gives us two answers, 𝑥 equal zero and 𝑥 minus six equal zero which follows through to give us 𝑥 equal six.

This means that the two solutions 𝑥 equal zero and 𝑥 equal six give undefined outputs and, therefore, will not be contained within the domain. We can go one step further by saying that the domain of 𝑛 of 𝑥 is all the real value minus the number zero and six.

We’re also required to simplify the function negative eight divided by 𝑥 minus six plus 𝑥 minus six divided by 𝑥 squared minus six 𝑥. In order to do this, our first step is to find a common denominator. Factorising out an 𝑥 on the denominator of the second term leaves us with 𝑥 minus six divided by 𝑥 multiplied by 𝑥 minus six.

Multiplying the top and bottom of the first term by 𝑥 gives us a common denominator of 𝑥 multiplied by 𝑥 minus six. As we now have a common denominator, we can write it as a single fraction: negative eight 𝑥 plus 𝑥 minus six divided by 𝑥 multiplied by 𝑥 minus six.

Grouping the like terms on the numerator, negative eight 𝑥 plus 𝑥 gives us negative seven 𝑥. And finally factorising out negative one gives us a simplified function of negative seven 𝑥 plus six divided by 𝑥 multiplied by 𝑥 minus six.

Therefore, the function negative eight divided by 𝑥 minus six plus 𝑥 minus six divided by 𝑥 squared minus six 𝑥 can be rewritten: negative seven 𝑥 plus six divided by 𝑥 multiplied by 𝑥 minus six with a domain of all real values with the exception of zero and six.