provide a detailed explanation of why a function must be a one-to-one function to have an inverse
Let's consider the properties of a function and one-to-one function.
f(x) is considered a function if for every value of x, there is one value of y. However, the values of y can be duplicated. So if f(x) contain the points (-2, 1) , (0, 5) , (2, 1), this is considered as a function.
A one-to-one function is a function in which for every x, there is exactly one y and for every y there is exactly one x. So for two or more values of x, the y's should have different values.
If f(x) contains the points (3,1) , (5, 2) and (7,1), it is a function, but not a one-to-one function.
If g(x) has the points (0,-2) (3, -4) and (6, -6), it is a one-to-one function.
To get the inverse of a function, the x and y are swap. However, the inverse can not always be a function. It can be a relation.
For example,`f(x) = x^2` is a function, but not a one-to-one function. Its inverse is
`f^(-1)(x) =+-sqrt(x). `
Since there is a plus and minus before the square root, it means that we have two inverse for the f(x). These are
`f^(-1)(x) =sqrtx` and ` f^(-1)(x) = -sqrt(x)`
Notice that for a single value of x, there are two values for y. So, the inverse of f(x) is not a function. It is only a relation.
Another example is f(x) = x + 1. This is a one-to-one function.
And its inverse is:
`f^(-1)(x) = x - 1`
Notice that for every x, there is only one value for y. Hence, the `f^(-1)(x)` is a function. Since it is the inverse of f(x) and it is a function,` f^(-1)(x)` is referred as inverse function.
Therefore, an inverse function` f^(-1)(x)` exists if f(x) is a one-to-one function.
Here is another way to think about why inverses only come out of one-to-one relationships:
For a curve to be considered a function, it needs to pass the "vertical line test" - in other words, a U-like parabola curve is a function, but a C-shaped curve is not. The inverse of a function is also a function and therefore has to satisfy this vertical line test as well.
The inverse of a function essentially takes the x-, y-coordinates and flips them. Graphically, this translates to flipping a line/curve across the line y = x. If you think about any vertical line (from the vertical line test) for the inverse, it must have come from a horizontal line of the original function (because the "vertical-ness" only comes after flipping the line across y = x). Therefore, you can now think of this as a horizontal line test on the original function.
If a function passes both the vertical line test and the horizontal line test (as we have shown above is required for a function to have an inverse), it is a one-to-one function. The vertical line test restricts any x from having multiple y-component possibilities, and the horizontal line test restricts any y from having multiple x-component possibilities. Therefore, each x must have only one y and each y can only have one x, making this a one-to-one function.