Square Root Function Breaking Rules?

Question

There is a rule that if a function is not one to one, then its inverse is not a function.

When graphed, a quadratic function is not one to one.

However, there is also a rule that the square root and radical sign with the default index of 2 only refer to the positive square root.

Is the square root function the inverse of a quadratic function?

If so, why are the quadratic and the square root functions special exceptions to this rule?

I have asked an accidentally duplicate question about why the square root symbol refers to only the positive one and it seems like it is only due to mathematical convention. Is it the same reason why?

Any help with explaining is greatly appreciated.

Answer 1

So you're right that there is a problem. The square root function is not an inverse to the function $f(x)=x^2$ on its domain. You can see that it doesn't satisfy the main thing that we want from an inverse: that $g(f(x))$ is equal to $x$. For example if we start with $-4$, and then put it first in $f(x)=x^2$ and then $g(x)=\sqrt{x}$, we get $\sqrt{(-4)^2} = \sqrt{16} = 4.$

The best that we can do here is take a branch of the function $f(x)=x^2$ where it does happen to be one-to-one. If we choose the set $[0,\infty)$ we then can take the inverse of that restricted function. That ends up being the function we know as $\sqrt{x}.$

This isn't that big of an issue so long as we are aware of what we are doing. The inverse trig functions such as $\arcsin(x)$ and $\arctan(x)$ all are inverses of a branch of the original function. For example if you take $\arctan(\tan(\pi))$ you will get $0$ instead of $\pi$.

There are some famous actual inverse functions in calculus though. $e^x$ and $\ln x$ are perhaps the most famous examples.

Answer 2

The function $f(x)=\sqrt{x}$ with domain $[0,\infty)$ is the inverse of the function $g(x)=x^2$ with domain $[0,\infty)$, i.e. a domain restricted function which equals the quadratic function on $[0,\infty)$ but is not defined for the negative real numbers.

If $g(x)=x^2$ were the quadratic function on $(-\infty,\infty)$ we would have $f\circ g(x)= \sqrt{x^2}=|x|$, and thus they are not inverse functions of each other.

Answer 3

The square root function is not the inverse of the squaring function, so there is no exception to the "rule".

Given a function $f: X \to Y$ and a function $g: Y\to X$, you say that $g$ is the inverse of $f$ if $f\circ g = Id_{Y}$ and $g\circ f = Id_{X}$. If $f$ is not one-to-one, an inverse cannot exist.

For example, consider the case $f:\mathbb{R}\to \mathbb{R}$ defined by $f(x) = x^2$ and $g:[0,\infty)\to [0,\infty)$ defined by $g(x) = \sqrt{x}$. Then, for example, $(g\circ f)(-3) = 3$, so $g$ is not the inverse of $f$.

You can however restrict the domain of the square function to the nonnegative real numbers, and then the square root function will be the inverse.

Answer 4

You have to be careful about the domain. You have probably heard of the vertical line test: if you have a purported function $f(x)$ then every vertical line must intersect the graph in one point. If it is less than one, so zero points, that value for $x$ is not in the domain. If it is more than one you don't have a unique function value.

For an inverse function this becomes the horizontal line test. Given a $y$ there needs to be only one $x$ such that $f(x)=y$. The parabola $f(x)=x^2$ fails this test because, for example, $3^2=9=(-3)^2$

If we restrict the domain of $f(x)$ to be $x \ge 0$ the graph passes the horizontal line test. Given a $y \ge 0$, which is the range of $f(x)=x^2$, there is only one $x \ge 0$ such that $x^2=y$. The square root function is the inverse of the square function if the domain of the square function is the nonnegative reals.

Answer 5

A function which is not 1 to 1 cannot have an inverse function. The square root function, whise range is the non-negative integers is not the inverse of any quadratic function defined on the real numbers. If we restrict the domain of f(x) = x^2 to the positive reals, then the square root function is its inverse. This is why the square root function is defined the way it is.