Video Transcript
Given the functions 𝑛 one of 𝑥 is equal to 𝑥 over 𝑥 squared minus 10𝑥 and 𝑛 two of 𝑥 is equal to one over 𝑥 minus 10, what is the set of values on which 𝑛 one is equal to 𝑛 two?
We’re given two rational functions 𝑛 one and 𝑛 two, where a rational function is a function of the form 𝑃 of 𝑥 over 𝑄 of 𝑥, such that 𝑃 and 𝑄 are polynomials in 𝑥. And the domain of the function, that’s the input values 𝑥, are such that 𝑄 of 𝑥 is not equal to zero. Now, we’re asked to find the set of values on which 𝑛 one is equal to 𝑛 two, that is, the domain or set of input values satisfying 𝑛 one equals 𝑛 two. This means we need to exclude any values of 𝑥 that make our denominators equal to zero. And so we need to solve 𝑥 squared minus 10𝑥 is equal to zero and 𝑥 minus 10 is equal to zero.
Let’s look first at 𝑛 one, where we have a common factor of 𝑥 in both terms. And taking this outside some parentheses, we have 𝑥 multiplied by 𝑥 minus 10 is equal to zero. Solutions to this are either 𝑥 is equal to zero or 𝑥 minus 10 is equal to zero. And to solve 𝑥 minus 10 is equal to zero, we add 10 to both sides, giving us 𝑥 is equal to 10. Our solutions are therefore 𝑥 is equal to zero and 𝑥 is equal to 10. The domain of 𝑛 one of 𝑥 is therefore the set of real numbers not including the set containing zero and 10.
And now turning to 𝑛 two of 𝑥, we see that the denominator is equal to zero if 𝑥 is equal to 10. And so the domain of 𝑛 two of 𝑥 is the set of real numbers not including the set containing the number 10.
Now let’s take a closer look at the function 𝑛 one of 𝑥. We’ve seen that we have a common factor of 𝑥 in the denominator, which we can take outside some parentheses. If we then divide both numerator and denominator by 𝑥, 𝑛 one is equal to one over 𝑥 minus 10, which is actually equal to our second function 𝑛 two of 𝑥. This means that the set of values on which 𝑛 one is equal to 𝑛 two is the domain of 𝑛 one. And this is because the domain of 𝑛 one includes the domain of 𝑛 two and our domain must apply to both functions. The set of values on which 𝑛 one is equal to 𝑛 two is therefore the set of real numbers not including the set containing the numbers zero and 10.
