1. [tex] y = \sqrt[3]{x-5} [/tex]
The inverse function can be found by interchanging x and y, then solving for y.
[tex] x = \sqrt[3]{y-5} [/tex]
[tex] x^{3} = y - 5 [/tex]
[tex] y = x^{3} + 5 [/tex]
This IS a function.
2. y = x^2 +4
.. x = y^2 +4
.. x -4 = y^2
.. y = ±√(x -4) . . . . . . the inverse relation is double-valued, so is NOT a function
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In general, odd degree polynomials and roots have inverses; even degree polynomials do not. Even-degree roots will typically be double-valued, so an inverse function can be defined for one or the other of the values, but not both.
In the above case,
.. y = √(x -4) is a function, applicable only for y ≥ 0.
