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I got stuck and thought if you could give some guidance on how to proceed?

find inverse to

$$y=\frac{\sqrt{x\:}+3}{\sqrt[5]{x}+4}$$

1 $$y=\frac{\sqrt{x\:}+3}{\sqrt[5]{x}+4}$$

2$$\:y\:\left(\sqrt[5]{x}+4\right)=\:\:\sqrt{x\:}+3$$

3 $$\:y\:\sqrt[5]{x}+4\:y=\:\:\sqrt{x\:}+3$$

4 $$y\:\sqrt[5]{x}\:-\:\sqrt{x\:}=\:\:+3\:-4y$$

JimmyK4542
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Math.emma
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  • Even Wolfram Alpha says nope http://www.wolframalpha.com/input/?i=inverse+of+y+%3D+(sqrt(x)+%2B3)%2F+(x%5E(1%2F5)%2B4) –  Sep 13 '18 at 17:10
  • If there is a typo and the denominator is really $5 \sqrt x + 4$ then there is no real problem. Similar for $\sqrt {5x} ; + 4$ – Will Jagy Sep 13 '18 at 17:16

1 Answers1

1

Let $x=t^{10}$, with $t\ge0$. Then you have $$ y=\frac{t^5+3}{t^2+4} $$ that becomes $$ t^5-yt^2+3-4y=0 $$

Then you can use How to solve fifth-degree equations by elliptic functions?, but there is no solution in “elementary functions”.

By the way, the function isn't invertible. Indeed, $$ f(t)=\frac{t^5+3}{t^2+4} $$ defined for $t\ge0$ has $$ f'(t)=\frac{5t^4(t^2+4)-2t(t^5+3)}{(t^2+4)}=\frac{3t^6+20t^4-6t}{(t^2+4)} $$ and the polynomial $3t^5+20t^3-6$ has a positive root.

egreg
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