Vtyjkyuk

I have this exercise;

For each of the following functions, determine the inverse function.

Here, $mathbbR_geq 0$ denotes the set of all non-negative reals:

$$f : mathbbR_geq 0tomathbbR, xmapsto sqrtx$$

But I really don't know where or how to start, could anyone provide some guidance on how to get an inverse function? maybe show some steps? Thank you so much

In general, if you have a function $y = f(x)$ and you want to find the inverse, then you want to rearrange $y = f(x)$ so that $x$ is a function of $y$.

For your example, observe that $y = sqrtx$ so that $y^2 = x$. Thus your inverse function is $f^-1(x) = x^2$.

It remains to determine the domain and range. We first note that $f : mathbbR_geq 0 to mathbbR$ has the domain $mathbbR_geq 0$ and the range $mathbbR$. However, the image of $f$ is the set of all elements in the range that are mapped to by $f$ from something in the domain. In this case, the image of $f$ is $mathbbR_geq 0$ (can you see why?). Thus, the domain of $f^-1$ will be this set, the image of $f$, and the range of $f^-1$ will be the domain of $f$ so that $f^-1 : mathbbR_geq 0 to mathbbR_geq 0$. Hence the inverse function is
$$
f^-1 : mathbbR_geq 0 to mathbbR_geq 0,quad f^-1(x) = x^2.
$$

It is important that the domain of $f^-1$ is the image of $f$ otherwise, as drhab points out, one may have something like
$$
-1stackrelf^-1to1stackrelfto1neq-1,
$$
which is not what we want since we should have $fcirc f^-1 = f^-1circ f = textIdentity map$.

We have $f(R)=R$ and $f$ is injective.

$y= sqrtx iff y^2=x$, hence $f^-1:R to R$ is given by $f^-1(x)=x^2$.

You can solve the question in the follwoing manner swap x and y then you will get a function $x=f(y)$ in terms of $y$ again swap x and y

$$y=sqrtx$$
$$y^2=x$$
$$x=y^2$$
$$f^-1(x)=x^2$$

The function $f:mathbb R_geq0tomathbb R$ prescribed by $xmapstosqrt x$ has no inverse.

This because it is not surjective.

The function $g:mathbb R_geq0tomathbb R_geq0$ prescribed by $xmapstosqrt x$ is bijective, hence has an inverse.

It is the function $mathbb R_geq0tomathbb R_geq0$ prescribed by $xmapsto x^2$.

For finding this function see the other answers.