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

Simplify the function 𝑛(𝑥) = 7𝑥²/(𝑥 − 1) + 3𝑥/(1 − 𝑥), and determine its domain.

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Video Transcript

Simplify the function 𝑛 of 𝑥 equals seven 𝑥 squared divided by 𝑥 minus one plus three 𝑥 divided by one minus 𝑥 and determine its domain.

In order to simplify the functions, we firstly need to find the lowest common denominator. In this case, we multiply 𝑥 minus one by one minus 𝑥. When multiplying fractions, we must do the same to the numerator as we do to the denominator.

Therefore, we need to multiply the numerator of the first fraction seven 𝑥 squared by one minus 𝑥 and the numerator of the second fraction three 𝑥 by 𝑥 minus one. This gives us a single fraction seven 𝑥 squared multiplied by one minus 𝑥 plus three 𝑥 multiplied by 𝑥 minus one divided by 𝑥 minus one multiplied by one minus 𝑥.

Our next step is to expand the brackets or parentheses using the distributive property. Seven 𝑥 squared multiplied by one is seven 𝑥 squared and seven 𝑥 squared multiplied by negative 𝑥 is negative seven 𝑥 cubed.

For the second bracket, three 𝑥 multiplied by 𝑥 is three 𝑥 squared and three 𝑥 multiplied by negative one is negative three 𝑥. Grouping the like terms simplifies the numerator to negative seven 𝑥 cubed plus 10 𝑥 squared minus three 𝑥. Our next step is to get the numerator in its simplest form by factorizing.

Well, firstly, we can see that an 𝑥 is common. Therefore, factorizing out the 𝑥 gives us 𝑥 multiplied by negative seven 𝑥 squared plus 10𝑥 minus three. The quadratic negative seven 𝑥 squared plus 10𝑥 minus three can be factorized into two brackets or parentheses. These two brackets are seven 𝑥 minus three and one minus 𝑥.

We could check this by using the FOIL method and expanding the bracket out. As one minus 𝑥 is a common term, we can divide the numerator and the denominator by one minus 𝑥. This leaves us 𝑥 multiplied by seven 𝑥 minus three divided by 𝑥 minus one. Therefore, the simplified version of 𝑛 of 𝑥 is 𝑥 multiplied by seven 𝑥 minus three divided by 𝑥 minus one.

The second part of the question asked us to determine the domain of the function 𝑛 of 𝑥. Well, at first glance, it appears that all real values are valid inputs to the function 𝑛 of 𝑥. However, on closer inspection, we can see that there is one value of 𝑥 that would make the denominator equal to zero. This would give us undefined value.

In order to calculate this value, we need to set the denominator — in this case 𝑥 minus one — equal to zero. Adding one to both sides of the equation gives us an answer for 𝑥 equal to one. This means that when we substitute 𝑥 equals one into the function 𝑛 of 𝑥, we get an undefined value.

This means that the domain of 𝑛 of 𝑥 is all the real values with the exception of one — the real values minus one.

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