Find the domain of the function 𝑓 of 𝑥 equals 𝑥 cubed minus 16𝑥 over 𝑥 squared minus four 𝑥.
Let’s begin by inspecting 𝑓 of 𝑥 in more detail. This is the quotient of a pair of polynomials. We’re dividing 𝑥 cubed minus 16𝑥 by 𝑥 squared minus four 𝑥. This means 𝑓 of 𝑥 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 𝑥 that make the denominator zero.
This means the domain of 𝑓 of 𝑥, then, will be the set of real numbers, excluding any values of 𝑥 that make 𝑥 squared minus four 𝑥 equal to zero. Since we don’t want the denominator to be equal to zero, we work out the values of 𝑥 that we need to exclude by setting it equal to zero and solving for 𝑥. That is 𝑥 squared minus four 𝑥 equals zero. To solve this equation, we’re going to factor the left-hand side. That gives us 𝑥 times 𝑥 minus four equals zero. Then for this expression to be equal to zero, either of its factors must be equal to zero: either 𝑥 equals zero or 𝑥 minus four equals zero.
Then we can solve this second equation by adding four to both sides, so either 𝑥 equals zero or 𝑥 equals four. These are the values of 𝑥 that make the denominator of 𝑓 of 𝑥 equal to zero. So, they’re the values of 𝑥 that we need to exclude from our domain. We can represent this in set notation, as shown.
The domain of the function 𝑓 of 𝑥 is the set of real numbers minus the set containing zero and four.