Find the domain of the real function 𝑓 of 𝑥 equals 𝑥 plus 48 over 𝑥 cubed minus 343.
Remember, the domain of a function is the set of possible inputs to that function. And in fact, inspecting 𝑓 of 𝑥, 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 𝑥 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 𝑥 is going to be the set of real numbers. But we want to ensure that the expression 𝑥 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 𝑥. This will tell us the value or values of 𝑥 that we can disregard from the domain of our function. So 𝑥 cubed minus 343 equals zero. To solve for 𝑥, let’s begin by adding 343 to both sides, so 𝑥 cubed is 343. Next, we’ll take the cube root of both sides. So 𝑥 is the cube root of 343.
In fact, the cube root of 343 is seven. So when 𝑥 is equal to seven, 𝑥 cubed minus 343 equals zero. So this is the value of 𝑥 we exclude from the domain of our function. The domain of 𝑓 of 𝑥 then is the set of real numbers minus the set containing the element seven.