### Video Transcript

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

Letβs begin by inspecting π of π₯ in more detail. This is the quotient of a pair of polynomials. Weβre dividing π₯ cubed minus 16π₯ by π₯ squared minus four π₯. This means π 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 π 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 donβt 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, weβre 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 π of π₯ equal to zero. So, theyβre 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 π of π₯ is the set of real numbers minus the set containing zero and four.