Question Video: Finding an Unknown in a Rational Function given Its Domain Mathematics • Third Year of Preparatory School

Video Transcript

Find 𝑘, given the domain of the function 𝑛 of 𝑥 equals negative seven over 𝑥 plus 𝑘 is the set of real numbers minus the set containing the element negative four.

Let’s begin by inspecting our function 𝑛 of 𝑥. It’s negative seven over 𝑥 plus 𝑘. In fact, it’s the quotient of a pair of polynomials, so we say it’s a rational function. And let’s think about what we know about the domain of a rational function. The domain of a rational function is just the set of all real numbers. But we have to exclude any values of 𝑥 that make the denominator of our function zero.

Now, we’re told that the domain of 𝑛 of 𝑥 is the set of real numbers, as expected, minus the set containing the element negative four. This must mean then that the value of 𝑥, negative four, must make the denominator 𝑥 plus 𝑘 equal to zero. We can therefore find the value of 𝑘 that also satisfies this equation by substituting 𝑥 equals negative four in. When we do, we get negative four plus 𝑘 equals zero. And then we can solve for 𝑘 by adding four to both sides. This means given that the domain of the function is the set of real numbers minus the set containing negative four, 𝑘 must be equal to four.