Video: Discussing the Continuity of Rational Functions

Find the set on which 𝑓(𝑥) = (𝑥 − 22)/(𝑥² − 2𝑥 − 63) is continuous.

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Video Transcript

Find the set on which 𝑓 of 𝑥 which is equal to 𝑥 minus 22 all over 𝑥 squared minus two 𝑥 minus 63 is continuous.

For this question, we have a rational function in the form 𝑃 of 𝑥 over 𝑄 of 𝑥. Now, we know that a rational function is continuous over its domain. And hence, our question reduces to finding the domain of our function 𝑓 of 𝑥. In essence, we want to find values of 𝑥 which would cause our function either to be undefined or to grow towards positive or negative infinity. Looking at the form of our function, we see that these troublesome points will occur when the denominator of our quotient, 𝑄 of 𝑥, evaluates to zero.

We can move forward with our question first by factorising 𝑄 of 𝑥. With a bit of inspection, we see that this quadratic factorises to 𝑥 minus nine times 𝑥 plus seven since these two numbers have a sum of negative two and a product of negative 63. From the factor theorem, we can then see that when 𝑥 is nine or when 𝑥 is negative seven, 𝑄 of 𝑥 will evaluate to zero. Putting this factorised version of 𝑄 of 𝑥 back into our function, we can then conclude that 𝑥 equals nine and 𝑥 equals negative seven are not in the domain of our function. This is because, at these values, the denominator of our quotient would be zero. And, hence, 𝑓 of 𝑥 would not give us a numerical evaluation.

Since 𝑓 of 𝑥 behaves at all other real values of 𝑥, we can say the following. The domain of our function 𝑓 of 𝑥 is the real numbers minus the set of nine and negative seven. And so, 𝑓 of 𝑥 is continuous over the real numbers minus the set of nine and negative seven. And here, we have answered our question.

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