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# Irrational Functions with Higher Indices by Christopher Monahan

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

. The exponent n in the polynomial function is called the index in the radical function. You know that
because 42 = 16. In the same way,
= 2 because 24 = 16. You also know that there are no real numbers for which x4 = −16, so
is not defined. This is consistent with what you saw in your graph of y = x4 and the restricted domain. Because −16 is not in the range of y = x4, −16 cannot be in the domain of
.

because 23 = 8, and
because (-2)3 = −8. In like manner,
because 35 = 243, and
because (-3)5 = −243.

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