Inverse Power Functions

If  y = f(x) = a x ^b, then we may solve for  x  in terms of  y  by taking roots:

x = f ^–1(y) = (y/a) ^1/b = (1/a) ^1/b y ^1/b = k y ^1/b ,

where  k = (1/a) ^1/b .

We see that the inverse of a power is another power. We often refer to a fractional power as a root. (For example, a  1/2  power is a "square root".) Thus the inverse of an integer power is a "root".

Saying when this inverse is defined takes some careful consideration. You may wish to review the variety of behaviors that are possible among power functions.

Clearly, neither  a  nor  b  may be  0 , for then either  1/a  or  1/b  will be undefined. In these cases, the forward power  f  degenerates to a constant function, with a graph that is a horizontal line.

There is another possible hitch to finding the inverse, however. If  b  is an even integer, or a fraction with an even numerator when in lowest terms, then we really should have written the following above:

x = f ^–1(y) = ±(y/a) ^1/b

There are actually two roots: both the positive and the negative values, when raised to the even  b^th  power, lead back to  x ^b = y/a  and hence to  a x ^b = y .