Question Video: Determining the Domain of a Rational Function Mathematics

Find the domain of the real function π(π₯) = (π₯ + 48)/(π₯Β³ β 343).

01:43

Video Transcript

Find the domain of the real function π of π₯ equals π₯ plus 48 over π₯ cubed minus 343.

Remember, the domain of a function is the set of possible inputs to that function. And in fact, inspecting π of π₯, we get a little bit more information from this. Itβs the quotient of two polynomials. This means itβs a rational function. So what do we know about the domain of a rational function?

The domain of a rational function is the set of real numbers. But we exclude any values of π₯ that make the denominator zero. And this is because we donβt want to be dividing by zero at any point. So, the domain of π of π₯ is going to be the set of real numbers. But we want to ensure that the expression π₯ cubed minus 343 is never equal to zero.

So, perhaps a little bit counterintuitively weβre going to set it equal to zero and solve for π₯. This will tell us the value or values of π₯ that we can disregard from the domain of our function. So π₯ cubed minus 343 equals zero. To solve for π₯, letβs begin by adding 343 to both sides, so π₯ cubed is 343. Next, weβll take the cube root of both sides. So π₯ is the cube root of 343.

In fact, the cube root of 343 is seven. So when π₯ is equal to seven, π₯ cubed minus 343 equals zero. So this is the value of π₯ we exclude from the domain of our function. The domain of π of π₯ then is the set of real numbers minus the set containing the element seven.