Determine the exact point(s) of intersection between f(x)=x^2-x-13 and it’s reciprocal function

Question

does anybody know how to solve this? I’m not sure what the reciprocal function means.

Answer 1

The reciprocal of a function $f(x)$ is simply $\frac{1}{f(x)}$. So let's denote it by $r(x)$ for convenience. Then, to find the point(s) of intersection, you want to find all $x$ that would satisfy $f(x) = r(x)$, or, written explicitly, $$x^2-x-13 = \frac{1}{x^2-x-13}.$$ Let's analyse this expression. When could it be that $a = 1/a$? Multiplying by $a$ we get $a^2=1$ which results in $a=\pm \sqrt{1}=\pm 1$. So the points of intersection of $f(x)$ and $r(x)$ are only when $f(x)=r(x) = \pm 1$. The goal now is to find the corresponding $x$-coordinates. In other words, you need to solve the two equations $$x^2-x-14 = 0, \\ x^2-x-12 = 0.$$ I trust you can take it from here?