Find the inverse of $f(x)=x^3+x+1$.

Question

$$f(x)=x^3+x+1$$ I didn't learn this at school and I want to know how I can get the inverse of this function.

Do you use differentiation?

I have this solution but I don't understand what it means $(f^{-1})'(3)$=$1$.

Is that differentiation?

Answer 1

First you have to check that $f$ is bijective (assuming that the domain and the codomain are both $\Bbb R$). The surjectivity is clear as $f$ is continuous and $\lim_{x\to\pm\infty}f(x)=\pm\infty$. For injectivity, we have $$f(x)=f(y)\iff x^3-y^3+x-y=0\iff (x-y)(x^2+xy+y^2+1)=0.$$ Since $x^2+xy+y^2+1=(x+y/2)^2+3(y/2)^2+1>0$, we see that $$f(x)=f(y) \iff x=y,$$ so $f$ is injective. Hence, $f$ is a bijection, and we can find the inverse.

Suppose that $f(x)=t$. Then, $x^3+x=t-1$. We assume alla Cardano's method that $x=u+v$ such that $x^2+1=u^2-uv+v^2$, so $u^3+v^3=x(x^2+1)=x^3+x=t-1$. Now, $$u^2-uv+v^2=x^2+1=(u+v)^2+1\implies uv=-\frac13\implies u^3v^3=-\frac1{27}.$$ That is, $u^3$ and $v^3$ are roots of $z^2-(t-1)z-\frac1{27}$, which means $$\{u^3,v^3\}=\left\{\frac{(t-1)\pm\sqrt{(t-1)^2+\frac{4}{27}}}{2}\right\}.$$ So, wlog, we can take $$u=\sqrt[3]{\frac{(t-1)+\sqrt{(t-1)^2+\frac{4}{27}}}{2}}\wedge v=\sqrt[3]{\frac{(t-1)-\sqrt{(t-1)^2+\frac{4}{27}}}{2}}. $$ This gives $$f^{-1}(t)=x=u+v=\sqrt[3]{\frac{(t-1)+\sqrt{(t-1)^2+\frac{4}{27}}}{2}}+\sqrt[3]{\frac{(t-1)-\sqrt{(t-1)^2+\frac{4}{27}}}{2}}.$$

Answer 2

If you are merely asked to compute $(f^{-1})'(3)$, then you don't need to find $f^{-1}$. It can be shown that $$(f^{-1})'(x)=\frac{1}{f'\big(f^{-1}(x)\big)}$$ whenever the denominator is nonzero. Since $f(1)=3$, we have $f^{-1}(3)=1$, and so $$(f^{-1})'(3)=\frac{1}{f'\big(f^{-1}(3)\big)}=\frac{1}{f'(1)}\,.$$ You should now be able to find $(f^{-1})'(3)$.

Since $f'(x)=3x^2+1$ for all $x\in\mathbb{R}$, we get $f'(1)=4$, so $(f^{-1})'(3)=\dfrac14$. This is not the same as what you claim (i.e., $(f^{-1})'(3)=1$). My answer can be verified by taking the derivative of Snookie's answer.

Answer 3

Switch $x$ and $y$ in $y=x^3+x+1$ to find the inverse function: $$x=y^3+y+1 \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$$ Take implicit derivative with respect to $x$: $$1=3y^2y'+y' \Rightarrow y'=\frac{1}{3y^2+1} \ \ \ \ \ \ \ (2)$$ For $x=3$, from $(1)$ we find $y(3)=1$. Hence: $$(y^{-1})'(3)=\frac{1}{3y^2+1}=\frac14.$$