What is the definition of a function being one-to-one?#neutrino said:
an output can have only one input ,but what i don't understand is the basis of the expressionMark44 said:
yes i can reach the conclusion if we draw a graph but I'm confused about how to arrive at the conclusion (one to one ) using this dataSimon Bridge said:
If you graph it, can you think of a specific feature of the graph that you could phrase mathematically?#neutrino said:
That is exactly what you said above: "an output can have only one input". If x and y are the "inputs" for the "outputs", f(x) and f(y), and they are the same, f(x)= f(y), so they are the same output, the inputs must be the same: x= y. Showing that "if f(x)= f(y) then x= y" is exactly the same as showing "an output can have only one input".#neutrino said:
It is the "If" part of an "if-then" statement. It is not proven, it is assumed. So there is no need for a "basis" for the statement.#neutrino said:
HallsofIvy said:
FactChecker said:
An inverse function is a function that "undoes" the action of another function. In other words, if a function maps an input x to an output y, the inverse function will take y as an input and return x as an output.
To prove that a function is its own inverse, you need to show that when you apply the function to its own inverse, you get back the original input. In other words, if the function is f(x), you need to show that f(f(x)) = x for all possible inputs x.
No, not all functions have an inverse. In order for a function to have an inverse, it must be one-to-one, meaning that each input has a unique output. Also, the function must be onto, meaning that every possible output has at least one corresponding input. If a function fails to meet these criteria, it will not have an inverse.
To prove that a function is one-to-one, you can use the horizontal line test. This test involves drawing horizontal lines across the graph of the function and checking if each line intersects the graph at only one point. To prove that a function is onto, you can use the vertical line test. This test involves drawing vertical lines across the graph of the function and checking if each line intersects the graph at least once.
Yes, the inverse of a one-to-one and onto function is always unique. This is because for every input, there is only one corresponding output, and for every output, there is only one corresponding input. Therefore, the inverse function will also be one-to-one and onto, and there can only be one such function.