Find the inverse function of ##f(x) =x^4+2x^2##

[YoungPhysicist]

Homework Statement

Find the inverse function of $f(x) =x^4+2x^2, x>0$

Homework Equations

$f(f^{-1}(x)) = x$

The Attempt at a Solution

My only progress so far is
$x^4+2x^2 = x^2(x^2+2)$
Then I am stuck.

Since my progress is close to nothing so I don’t expect a complete explanation. Sorry!

A link to some useful website talking about this will be greatly appreciated since there are actually a dozen of similar problems that I can’t solve. Just kind of want to get this once in for all, and not just this particular one.

Thanks again!

 

[PeroK]

Hint: is there a quadratic equation hiding somewhere?

 

[Delta2]

You have to set $f(x)=y$ and then try to solve the equation for $x=f^{-1}(y)$, consider y as known. In this case you have to solve the equation

$x^4+2x^2-y=0$ which is a bi-quadratic equation

 

[PeroK]

$x^4+x^2-y=0$ which is a bi-quadratic equation

There it is!

 

[YoungPhysicist]

You have to set $f(x)=y$ and then try to solve the equation for $x=f^{-1}(y)$, consider y as known. In this case you have to solve the equation

$x^4+2x^2-y=0$ which is a bi-quadratic equation

Sorry, do you mean getting all the terms to the other side leaving only x=?

 

[YoungPhysicist]

And by the way, the problem did mention $x>0$.

 

[Wrichik Basu]

Sorry, do you mean getting all the terms to the other side leaving only x=?

Yes. Just find $x$ in terms of $y$.

 

[Delta2]

You might find useful to set $z=x^2$ firstly and solve the equation $z^2+2z-y=0$ first, then take $x=\sqrt{z}>0$

 

[Mark44]

At heart, you're solving for x in the equation $y = x^4 + 2x^2$. Notice that if you add 1 to $x^4 + 2x^2$ it's a perfect square.

 

[Delta2]

Because haven't said it explicitly yet, for similar problems where you are given $f(x)$ and need to find the inverse of $f$, all you need to do is solve the equation $f(x)=y$. By solving it you ll have found x in terms of y, or in other words your ll find $x$ as a function $g(y)$ . The function $g$ is the inverse of $f$ because we know for the inverse of $f$ it is
$$f(x)=y\iff x=f^{-1}(y)$$. (1)

But because $x=g(y)$ is the solution of the equation $f(x)=y$ we also have

$$f(x)=y\iff x=g(y)$$ (2)

From (1) and (2) we can conclude that $g=f^{-1}$

 

[Wrichik Basu]

In addition to what @Delta2 has said in post #10, keep in mind that not all functions have an inverse. Although books almost always give functions that have an inverse, when you apply this concept to other subjects, be sure to check whether the function is bijective. I believe this has already been written in your book.

 

[YoungPhysicist]

At heart, you're solving for x in the equation $y = x^4 + 2x^2$. Notice that if you add 1 to $x^4 + 2x^2$ it's a perfect square.

Yes. Just find $x$ in terms of $y$.

Ok. According to those facts, here is what I came up with:
$$
y = x^4+2x^2\\
\Rightarrow y+1 = x^4+2x+1 = (x^2+1)^2\\
\Rightarrow x^2+1 = \sqrt{y+1}\\
\Rightarrow x = \sqrt{\sqrt{y+1}-1}$$
Which according to my books answer(no process at all) is correct. Thanks everyone!