Question Video: Determining the Domain of a Rational Function Mathematics

Find the domain of the real function 𝑓(π‘₯) = (π‘₯ + 48)/(π‘₯Β³ βˆ’ 343).

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Video Transcript

Find the domain of the real function 𝑓 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 𝑓 of π‘₯, we get a little bit more information from this. It’s the quotient of two polynomials. This means it’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 don’t want to be dividing by zero at any point. So, the domain of 𝑓 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 we’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‘₯, let’s begin by adding 343 to both sides, so π‘₯ cubed is 343. Next, we’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 𝑓 of π‘₯ then is the set of real numbers minus the set containing the element seven.

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