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Question Video: Finding the Domain of Rational Functions Mathematics

Find the domain of the function 𝑓(π‘₯) = (π‘₯Β³ βˆ’ 16π‘₯)/(π‘₯Β² βˆ’ 4π‘₯).

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Video Transcript

Find the domain of the function 𝑓 of π‘₯ equals π‘₯ cubed minus 16π‘₯ over π‘₯ squared minus four π‘₯.

Let’s begin by inspecting 𝑓 of π‘₯ in more detail. This is the quotient of a pair of polynomials. We’re dividing π‘₯ cubed minus 16π‘₯ by π‘₯ squared minus four π‘₯. This means 𝑓 of π‘₯ is, in fact, a rational function. A rational function is the quotient of a pair of polynomials. So, what do we know about the domain of a rational function? The domain of a rational function is the set of all real numbers, not including any values of π‘₯ that make the denominator zero.

This means the domain of 𝑓 of π‘₯, then, will be the set of real numbers, excluding any values of π‘₯ that make π‘₯ squared minus four π‘₯ equal to zero. Since we don’t want the denominator to be equal to zero, we work out the values of π‘₯ that we need to exclude by setting it equal to zero and solving for π‘₯. That is π‘₯ squared minus four π‘₯ equals zero. To solve this equation, we’re going to factor the left-hand side. That gives us π‘₯ times π‘₯ minus four equals zero. Then for this expression to be equal to zero, either of its factors must be equal to zero: either π‘₯ equals zero or π‘₯ minus four equals zero.

Then we can solve this second equation by adding four to both sides, so either π‘₯ equals zero or π‘₯ equals four. These are the values of π‘₯ that make the denominator of 𝑓 of π‘₯ equal to zero. So, they’re the values of π‘₯ that we need to exclude from our domain. We can represent this in set notation, as shown.

The domain of the function 𝑓 of π‘₯ is the set of real numbers minus the set containing zero and four.

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