Given that 𝑛 sub one of 𝑥 is equal to seven plus 𝑏 over 𝑥 minus seven, 𝑛 sub two of 𝑥 is equal to two over 𝑥 minus seven, and 𝑛 sub one of 𝑥 is equal to 𝑛 sub two of 𝑥, what is the value of 𝑏?
Here, we have two rational functions. And we’re told that the two functions are equal to each other. So that means that for 𝑛 sub one of 𝑥 to be called 𝑛 sub two of 𝑥, the functions are equal for all values of 𝑥, not just for a certain value or values. So we can say that seven plus 𝑏 over 𝑥 minus seven equals two over 𝑥 minus seven. Now, look really carefully. The denominator of each of our functions is equal. And so for the function 𝑛 sub one of 𝑥 to be equal to the function 𝑛 sub two of 𝑥, their numerator must also be equal. So seven plus 𝑏 must be equal to two.
Now, we have this; we can solve this equation for 𝑏 fairly easily. We’re going to subtract seven from both sides. And seven plus 𝑏 minus seven is 𝑏. And two minus seven is negative five. And so we found the value of 𝑏; it’s negative five. Now, of course, we can check that what we’ve done is correct by substituting 𝑏 into our expression for the function 𝑛 sub one of 𝑥. We get seven plus negative five over 𝑥 minus seven, which is two over 𝑥 minus seven as required.