Hello! I know that a function does not have an inverse if it is not a one-to-one function, but I don't know how to prove a function is not one-to-one. Please teach me how to do so using the example below! Thank you!
The function f is defined as f(x) = x^2 -2x -1, x is a real number. Show that the inverse function of f does not exist.
first read this:
Injective function - Wikipedia, the free encyclopedia
now you can show for example that this quadratic polynomial has two distinct roots:
so the function is not 1:1
Well, using only f(x) = 0 is not enough for every quadratic. You can use a plotter and look for x values that give the same f(x). Or in this situation you can,
For and it doesn't have a real root.
For and it has 2 equivalent roots. (or you can say that it has only one root)
For and it has 2 different real roots.
Thus, any a value greater than -2 will make have 2 values.
A function is 1-to-1 if it also satisfies the horizontal line test - you can run a horizontal line from down to up and the horizontal line will never cut the fuction more than once.
For your example, f(x) = x^2 -2x -1, run a horizontal line from down to up. After the horizontal line gets past the turning point, it cuts the parabola TWICE - that's more than once!! So f(x) = x^2 -2x -1 is NOT a 1-to-1 function.
If you restricted the domain of f(x) = x^2 -2x -1 = (x - 1)^2 - 2 to be x > 1, say, then the horizontal line will never cut more than once and so the function is now 1-to-1. Can you think of another restriction on the domain to make f(x) = x^2 -2x -1 a 1-to-1 function?
if you are allowed to use drivative you can use the property that with a one-to-one-function the sign of the first derivative doesn't change and the first derivative is unequal zero.
With your example:
has a change of sign from - to + at x = 1
and therefore f can't be a one-to-one-function.
Take as an eaxample the function and it's inverse function .
If the "behaviour" of a function changes from decreasing to increasing (example then the sign of the gradient changes from - to +. With a differentiable function the first derivative will be zero at the point where the monotony changes. This point (turning point?) is a minimum point (in this case).
The function isn't differentiable at x = 0. But the gradient changes from -1 to +1 at x = 0 and therefore you have a minimum point at x =0.
If the monotony changes the function has at least 2 "branches" which will be intersected by a horizontal line twice. Such a function has no inverse function because the x-coordinates of the intersection points must be the y-coordinates of the inverse function and that's opposing the definition of a function.