Irrational Functions with Higher Indices
The square root function is the inverse of the polynomial y = x2. All of the polynomials whose equation is y = xn, where n is a positive integer, have inverses. Take a moment to use your graphing calculator to look at the graphs of y = x4, y = x6, and y = x8. Do you see that they are very similar to the graph of y = x2? They are symmetric to the y-axis, they pass through the origin, and they never drop below the x-axis. Because they fail the horizontal-line test for 1-1 functions, you must restrict the domain to be x > 0 in order to have an inverse function.
Take a look at the graphs of y = x3, y = x5, and y = x7. These graphs are not symmetric to the y-axis, they pass through the origin, they do have output values that are negative, and they do pass the horizontal-line test. These functions do have inverses as they exist, so the domain does not have to be restricted.
The domain of a radical function is x > 0 when the index is even and is all real numbers when the index is odd.
The inverse of the function f(x) = xn is