Evaluate a Variable Defined in Terms of its Function

Question

I have a variable x which is defined as follows:

x = 150 / (7 + f(x)) where f(x) = 0.005 * x if x > 200, or 100 otherwise.

This is actually a simplified version of a real world problem.

How do I evaluate x?

Answer 1

You have 2 cases. If $x \leq 200$ then $f(x) = 100$ and your definition says $$ x = \frac{150}{7+f(x)} = \frac{150}{107} < 200 $$ which seems to be one of the solutions.

Otherwise, $x > 200$ and $f(x) = 0.005x$ and your equation defines $x = \frac{150}{7+0.005x}$, which is equivalent to $$ 0.005x^2 + 7x - 150 = 0, $$ which implies either $x = 100 \left(2\sqrt{13}-7\right) \approx 21.11 < 200$ (which is impossible since we are in the case $x > 200$) or $x = -100\left(7+2\sqrt{13}\right) < 0$ which is impossible for the same reason.

This leaves only one solution possible, $x = \frac{150}{107}$.