Video: Simplifying a Rational Function and Determining Its Domain

Rhodri Jones

Simplify the function 𝑛(𝑥) = 1/(𝑥 + 3) − 8/(𝑥 + 3), and determine its domain.


Video Transcript

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

When simplifying two fractions, we need to ensure that we have a common denominator. In this case, the denominator of both terms is 𝑥 plus three. So we can write the function as a single fraction. One divided by 𝑥 plus three minus eight divided by 𝑥 plus three can be rewritten: one minus eight divided by 𝑥 plus three.

As one minus eight is equal to negative seven, the function in its simplest form can be written negative seven divided by 𝑥 plus three. We were also asked to determine the domain of the function. Now at first glance, it appears that we can put in all real values into our function.

However, when look more closely, there is a value for 𝑥 that would make the denominator equal to zero, which would give us undefined values. In order to find this value, we need to set the denominator, 𝑥 plus three, equal to zero. Subtracting three from both sides of this equation gives us a value of 𝑥 of negative three.

This means that when we input negative three into the equation, we end up with an undefined output. Therefore, negative three cannot be contained within the domain. This means that the domain of the function 𝑛 of 𝑥 is all real values with the exception of negative three.

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