Cross Examination

Algebra Level pending

Let's say we have a function $f(x)$ . Now let's say we have another function $g(x)$ . $g(x)$ is defined as being $\frac{1}{f(x)}$ . Assume that $f(x)$ does not have any vertical shift (such as $y = x + 2$ ).

There are two y-values where $f$ and $g$ cross, sometimes more than once. Where are they?

1, 0.5 -1, 0 1, 0 -1, 0.5 1, -1

1 solution

Jacob Day
Oct 2, 2017

Assuming that $f$ has no vertical shift as stated in the problem, the answer is 1 and -1 for most cases. I haven't found a counterexample yet.

This can be proved simply. Assume the equation $x = \frac{1}{x}$ . Multiply $x$ on both sides to get $x^2 = 1$ . Solving this gives 1 and -1. It is important to note that both of these values fit the equation.

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