I am currently studying functions in general, and I've come across left and right inverses, however I can't wrap my head around the following: $f(x) = 3x^4$

I know this function doesn't have an inverse because we get the quad root of $x/3$, however does it have a left/right inverse?


This function does not have an inverse function, however, the inverse of the image $f(x) = 3 x^4$ can be found by piecing together two functions:

I'll use $f^{-1}(x)$ to denote the this inverse image:

$$f^{-1}(x) =\pm \dfrac{\sqrt[\large4]x}{\sqrt[\large4]3}$$

and "left or right" is irrelevant.

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The portion of the graph of $f^{-1}$ at and above the $x$-axis is given by $\left(+ \dfrac{\sqrt[\large4]x}{\sqrt[\large4]3}\right)$.

The rest is given by $\left(-\dfrac{\sqrt[\large4]x}{\sqrt[\large4]3}\right)$.

  • $\begingroup$ Okay, that clarified some things for me, but I am still a little unclear of what right/left inverse mean, I've been searching up on it and I can't really find any documentation that I can understand it properly from. Thanks btw :) $\endgroup$ – Frows Jan 7 '14 at 18:41
  • $\begingroup$ If an inverse exists, it is both left and right: $(f\circ f^{-1})(x) = (f^{-1}\circ f)(x) = x$, as we define what the inverse of a function is. Only Left, I'm assuming means that only $(f^{-1}\circ f)(x) = x$, only right means only $(f\circ f^{-1})(x) = x$. $\endgroup$ – Namaste Jan 7 '14 at 18:42
  • $\begingroup$ Oh okay, and what if I had two different functions in a composition, rather than the same that present identity? $\endgroup$ – Frows Jan 7 '14 at 18:45
  • $\begingroup$ Strictly speaking, the inverse of a function is unique: if you find a function such that $(f\circ f^{-1})(x) = (f^{-1}\circ f)(x) = x$, then $f^{-1}$ is unique: no other distinct function will satisfy this property. $\endgroup$ – Namaste Jan 7 '14 at 18:49
  • $\begingroup$ It has, also I should probably clarify abit more what I mean by two different functions, I meant having for example f(x) = x+1 and g(x) = x-2, if I were to find an inverse of a composition of g(f(x)), perhaps not in this particular case, but in a general case where a composition has a left inverse, but doesn't have a right inverse and vice versa, how would I know which one it is? $\endgroup$ – Frows Jan 7 '14 at 18:54

A surjective function has a right inverse. So if your function is defined as $$ f\colon \mathbb R \to [0,+\infty)\\ f(x) = 3x^4 $$ it has a right inverse $$ g\colon [0,+\infty) \to \mathbb R\\ g(y) = \sqrt[4]{y/3} $$ in fact $f(g(y)) = y$ for all $y\ge 0$.

If your function is defined as $f\colon \mathbb R\to \mathbb R$ it has neither right nor left inverse because it is neither surjective nor injective.


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