Undoing a Conversion

A conversion rule turns one unit into another, and its inverse turns it back. In the US, a man's shoe size is about 3 times his foot length in inches, minus 22: $f\left(x\right) = 3x-22$. A 10 inch foot takes size 8. To go from a size back to a length, you need ${f}^{-1}\left(x\right)$.

The method is the one from the inverse chapter. Write the rule as $y = 3x-22$, swap x and y, then solve for y. Swapping gives $x = 3y-22$. Add 22 to get $x+22 = 3y$, then divide by 3: $y = \frac{x+22}{3}$. Check: size 8 gives 30 over 3, which is 10 inches, the length we started from.

Why swap? The inverse takes the old output as its input, and we keep the letter x for an input. When the rule multiplies by a fraction, divide by that fraction, which is the same as multiplying by its flip: $x-4 = \frac{2}{3}y$ gives $y = \frac{3}{2}\left(x-4\right)$.