Simple way to calculate the inverses of these functions

Question

if we have the following functions:

$f(x)=x^3+x$

$g(x)=x^3+x-9$

$h(x)=x|x|+1$

is there a simple way to calculate these inverses:

$f^{-1}(2)$

$g^{-1}(1)$

$h^{-1}(3)$

Answer 1

Note that since $f(x)=x^3+x$ is injective since $f'(x)=x^2+1>0$

• $f(x)=x^3+x=2\implies x=1 \implies f^{-1}(2)=1$

and also for $g(x)=x^3+x-9$ injective since $g'(x)=x^2+1>0$

• $g(x)=x^3+x-9=1\implies x=2 \implies g^{-1}(1)=2$

and also for $h(x)=x|x|+1$ injective (can you say why?) we can find

• $h(x)=x|x|+1=3\implies x=\sqrt 2 \implies h^{-1}(3)=\sqrt2$

Answer 2

For $$h(x)=x|x|+1$$ it is not hard to find the inverse function, we only need $h^{-1}(x)$ for $x\geq 0$, so we have $$h^{-1}(x)=\sqrt{x-1}$$