Inverse function, Injective functions, Domain, Range, Find the inverse.
Each function is a assignment, the inverse of which, if it exists at all, is the reverse assignment.
Calculating the inverse goes like this:
First, write the function in the form of y=thingy:
Here, we assign y to x.
The inverse is the reverse assignment, where we assign x to y. So, the purpose is always to rearrange y=thingy to x=something.
Finally, we swap x and y (some people don’t do this), and then we get the inverse.
The inverse is denoted by:
But, there is a little trouble. Not all functions have an inverse, as not all assignments can be reversed.
For example, in the case of , we have and , and thus, we cannot reverse this: .
The crux of the problem is that this function assigns the same number to two different numbers (2 and -2), and therefore, the assignment cannot be reversed.
But if we exclude the negative numbers,
then everything will be all right.
So, only those functions have an inverse where for two different x values
there are two different y-s assigned.
We say they have a one-to-one correspondence, shortly, they are injective.
A function is injective, if then .
All strictly monotonic functions are injective and thus, invertible.
And there is one more thing here.
Let the function be and its domain .
Well, then the image is .
The inverse function is the reverse assignment, so in this case, these are swapped.
If is invertible, then its Domain is identical to the Range of its inverse, and its Range is identical to the Domain of the inverse.
Let's see a few examples!
Let’s find the inverse of function , if
There is no inverse, because the function is not injective.
For example, it assigns the same number to 4 and -4, namely 0.
This case is entirely different, as x could only be positive here. There are no two positive numbers with the same square, so this function is injective.
Let’s see the inverse:
In this case, there is an inverse, too, because the function is injective.
Let’s see the inverse!
In this case the function has no inverse, because it is - again- not injective. For example, it assigns the same number to 4 and -4, namely 0.
Unfortunately there is no inverse in this case either, as the function is not injective.
Let’s see one more.
Here is this function, and we want to find its inverse.
Finally let’s see this one, too:
Let’s talk a bit about the geometric meaning of the inverse.
Here is a function
and let’s see what happens to the graph of this function when we invert it.
Let’s reflect the graph of the function about the y=x line.
It is clearly visible on the drawing that the inverse of radical functions is never a full parabola, only a half.
And the reverse is true as well: a full parabola can never be inverted, only its half.
Here comes another splendid function:
Well, the inverse of this function is:
The inverse of the exponential functions are the logarithmic functions.
And this is mutual: the inverse of the logarithmic functions are the exponential functions.
Let’s see the inverse of this, for instance:
We can lure the x out of the exponent by taking the logarithm of both sides.
Here is another one, for example:
Functions and are also inverses of each other.
We should be careful with this function inversion thing, as it could be harmful in large doses.
But maybe we can get away with one more...