Inverse functions

An inverse function is one that assigns elements "in the opposite way" from the original function.

Introductory example

To begin, consider a simple linear function f(x) = 2x. What will the function values of this function look like? For example, for the first few natural numbers, the function values will look like this:

$$\begin{eqnarray} f(1)&=&2\\ f(2)&=&4\\ f(3)&=&6\\ f(4)&=&8\\ &\ldots& \end{eqnarray}$$

We can see that if we put a two in the function, it will return a four. What should the inverse function to this function do? It should work the other way around - we should put a four into it and the function should return a two. The inverse function, denoted by f⁻¹, should display the elements in reverse, i.e. like this:

$$\begin{eqnarray} f^{-1}(2)&=&1\\ f^{-1}(4)&=&2\\ f^{-1}(6)&=&3\\ f^{-1}(8)&=&4\\ &\ldots& \end{eqnarray}$$

How to do this? If the original function returned twice the parameter x, then the inverse function should in turn return half the parameter x. If we put in a three, we have a six after doubling. If we take half of the six, we get back a three. The inverse function would then have the following notation:

$$f^{-1}(x)=\frac{x}{2}$$

Definition

Now let's define the inverse function properly. So let's have a function f with definition scope D(f) and value scope H(f). From the definition of the function, for all elements x of the definition scope D(f), we have some element y of the value scope H(f), for which f(x) = y holds.

The inverse function f⁻¹ is then the function for which it holds:

$$f(x)=y\Leftrightarrow f^{-1}(y)=x.$$

What does this definition tell us? That if we call the function f with the argument x and get the value y, then the inverse function must be called with the argument y and must return the value x.

We'll show all this in the figures. The first figure represents the original function f, which returns twice the passed value. The left circle represents the definition scope, the set from which we select values for x. The right circle represents the value scope, the set of values that the function value can take. The arrows then indicate which input is displayed to which output. Of course there should be infinitely many arrows, this is just a small part of the function.

The following is a picture of the inverse of the function. The inverse function will look very similar, in particular the direction of the arrows will change and the definition scope and value scope will change. These are also swapped.

We can see that the original function displayed from D(f) to H(f), but the inverse function displays the opposite, from H(f) to D(f). Thus, the definitional domain of the original function is equal to the value domain of the inverse function, and the value domain of the original function is equal to the definitional domain of the inverse function.

The existence of the inverse function

First of all, the inverse function may not always exist because of a fundamental property of the function. If we take two identical elements of the definitional domain a = b, then the function must always return the same result f(a) = f(b). If we go back to the function that returned twice, then it must always be true that f(5) = 10. It cannot be that the function returns, for example, thirteen: f(5) = 13 - this must not occur for a function, by definition.

Now let's try to analyze the situation where we have, for example, a quadratic function f(x) = x². What will the function return if we call it with the arguments x₁ = 2 and x₂ = −2? The function will return the value 4 in both cases:

$$\begin{eqnarray} f(2)&=&4\\ f(-2)&=&4 \end{eqnarray}$$

OK then. But what would the inverse function look like? If we were to construct an inverse function to this quadratic function, it would also have to hold:

$$\begin{eqnarray} f^{-1}(4)&=&2\\ f^{-1}(4)&=&-2 \end{eqnarray}$$

And we know that is not possible. The function cannot return different results for the same arguments. So when does an inverse function exist? If for no two input arguments does the function return the same value. Which is exactly the definition of a simple function:

$$x_1\ne x_2\Rightarrow f(x_1)\ne f(x_2)$$

An inverse function to the function f exists precisely when the function f is simple. Let's use the figures to explain why there really can't be an inverse function to a function that is not simple. Let's use the arrows to show part of the quadratic function f(x) = x²:

If we tried to invert the arrows, we would get a picture like this:

We get a picture where two arrows lead from one point to different numbers in the other set. This does not meet the definition of a function, so the inverse function cannot exist.

Graphical meaning

If you take the graph of a function, then if there is an inverse function, then these graphs will be axisymmetric with the first and third quadrants - which also represents the graph of the function f(x) = x.

Calculating the inverse function

If you are given a function and the task is to calculate the inverse function to it, the procedure is to build the function into the equation and isolate it from the equation x. So we modify the first linear function f(x) = 2x as follows:

$$y=2x$$

This is how we start. The goal is to express x relative to y. Here it goes pretty quickly, you just divide the equation by two

$$\frac{y}{2}=x$$

and you have the inverse of x expressed as $f^{-1}(x)=\frac{x}{2}$. Another example:

$$f(x)=4x-7$$

Substitute x into the equation and isolate :

$$\begin{eqnarray} y&=&4x-7\quad/+7\\ y+7&=&4x\quad/:4\\ \frac{y+7}{4}&=&x \end{eqnarray}$$

The inverse function will look like this:

$$f^{-1}(x)=\frac{x+7}{4}$$

We can try it out. We calculate the functional value of the original function with value three:

$$f(3)=4\cdot3-7=12-7=5$$

If we calculated the inverse function correctly, we should get back three when called with a value of 5:

$$f^{-1}(5)=\frac{5+7}{4}=\frac{12}{4}=3$$

Another example:

$$f(x)=x^2+1$$

Let's isolate x:

$$\begin{eqnarray} y&=&x^2+1\quad/-1\\ y-1&=&x^2\quad /\sqrt{}\\ \sqrt{y-1}&=&\sqrt{x^2} \end{eqnarray}$$

Now we are in an awkward situation. x² because we can't just square root x, we have to add an absolute value, i.e.:

$$\sqrt{x^2}=|x|$$

If we add it to the previous equation:

$$\sqrt{y-1}=|x|$$

But this gives us two results, one positive and one negative:

$$\begin{eqnarray} x_1&=&\sqrt{y-1}\\ x_2&=&-\sqrt{y-1} \end{eqnarray}$$

The result is that for one x we have two different y, except when the square root will be zero. However, we can find the inverse function on a certain interval. If we look at the graph of the function:

we find that the inverse function could exist on the interval <0,∞). So we can just take the isolation for x₁. At this point we get the function whose graph you see in the figure:

So we can say that the inverse function exists on the interval

$$f^{-1}(x)=\sqrt{x-1}$$