A function simply gives you an output depending on the input given. In functions, a value is provided and the function performs some operations on it to give an answer. For example, the function f(x) = x + 1 adds 1 to any value that you give it. If you give it a 5, the function will give you a 6:f(5) = 5 + 1 = 6. Functions have requirements that they must meet, though. It can be the x value or the input and they cannot be linked to more than one answer. This simply means that you cannot give a function one value and it gives you two different answers. In a one to one function, there is a rule that gives a correspondence between elements in two given sets. Domain and range are used such that every element in a domain corresponds to only one element in the range. A one to one kind of function is a function in which each element in a range of that function corresponds to one element of a domain. A one to one function is often expressed as 1-1. It is important to note that y=f(x) is a function only when is passes the vertical line test. For a function to qualify to be a one to one function:

- It has to pass both the horizontal line test and the vertical line test. The horizontal line test states that a graph represents a one to one function if every horizontal line intersects that graph only once and the vertical line test states that a function should have only one output y for every unique input x.
- No two different elements in the domain should have the same element in the range.

One to one kind of functions are represented algebraically as x1 and x2. The x1 and x2 are any elements of the domain and a function is represented by f(x). f(x) is a one to one function if x1 is not equal to x2 which means f(x1) is also not equal to f(x2). Or if f(x1) is equal to f(x2) then x1 is equal to x2. One to one functions preserve distinctness in that they will never map distinct domain elements to the elements of their range. A function relates every value of the variable x input to a single value of the dependent variable y which is the output. It is possible to have two or more inputs giving a similar output. Consider the function y = x 3 − 6x 2 + 11x − 6. Using simple substitution, the inputs x = 1, x = 2 and x = 3 give the same output y = 0. In this particular case, three values of x are related to one value of y which means that it is a three to one function and not a one to one kind of function. A function in which the same output never repeats for different inputs is a one to one function.

In a one to one function, any y given has only one x that can be paired with it. The one to one kind of functions are also known as injectives. If f(x) = x³ is a one to one function, this cubic function implies that every x value has a one unique y value that is not used by the other x elements. This is a characteristic of a one to one function. Let’s compare {(2, 3),(4, 5),(1, 5),(3, 4)} and {(2, 3),(4, 2),(1, 5),(3, 4)}. The first function has (4,5) and (1,5) which implies that the inputs 4 and 1 give a similar output 5. The first function here is not a one to one kind of function. There is no repetition of output on the second function which means the second function is a one to one function.

If you are asked to compare a function y=x2 with the function y=3x+1, the first function repeats the output of y=4 for the inputs x = 2 and x = −2 (4 = 22 and 4 = (−2)2. The function is a one to one type. The outputs of the second function are not repeated which implies that the function is a one to one function. In the horizontal line test, a horizontal line has all points which have their y coordinates equal to the same number. If the y coordinates are equal to a number 2 then the line can be described in an equation as y=2 which implies that every value of x is related to 2. Going by the definition of a one to one function, at most one x value is related to a given value of y. It then follows that a horizontal line intersects the graph of a function once. A graph of a function on a coordinate plane is a graph of a one to one function if there is no horizontal line intersecting the graph more than once.

One to one functions can also be inverse functions. If f is a one to one type of function with a domain A and a range B, its inverse function will be represented as f-1 and will have a domain B and a range A. It is defined by f −1 (y) = x only if f(x) = y for any y in B. To get the inverse of a one to one function you begin by replacing the function f(x) with y. 2. The next step is to interchange x and solve the equation for y. The final equation will be f −1 (x). If f is a one to one function with domain A and range B and its inverse function satisfies f −1 (f(x)) = x for each x in A and f(f −1 (x)) = x for each x in B, then the inverse of f −1 is f. An inverse function interchanges the domain and the range. The domain of f = Range of f −1 and the range of f = domain of f −1. A graph of an inverse function is gotten by reflecting the graph of f across the line y=x and it is only the one to one functions that can have an inverse.

You can determine if functions are of the one to one kinds graphically by using the horizontal line test which states that a graph represents a one to one function if every horizontal line intersects that graph only once. But you cannot proof that a function is a one to one function by just looking at the graph. This is because a graph is a small portion of a function and you generally need to proof that it is one to one function on the whole domain. There are two main methods of verifying that a function is a one to one function.

The first method is showing if f(x1) is equal to f(x2) then x1 is equal to x2. This simply means that if a function has the same value at two points then those points must be equal. In other words, if a function contains the same value at two points, those points can never be different. This statement is simply what the horizontal line test states. A graph is a one to one function when there are no two x values which are assigned to the same y value. If you are asked to verify that f(x)=1 is a one to one function, you will begin by assuming that there exists some x1 and x2 in that f(x1)=f(x2). This means that 1x1=1x2 but 1x1=1x2⇒1x1−1x2=0⇒x1−x2x1x2=0⇒ the numerator should be equal to zero. For example, x2−x1=0⇒x2=x1.

The second method of verifying that a function is a one to one kind is by showing that a function is always decreasing or always increasing. And it will always pass the horizontal line test. A function f increases its domain whenever x2 is greater than x1. This means that f(x2) is greater than f(x1). As x becomes bigger f(x) also grows bigger. A function f decreases on its domain whenever x2 is greater than x1. This means f(x1) is bigger than f(x2). As x grows bigger, f(x) becomes smaller.

To verify that y=9-x2 is a one to one function you will first need to solve the equation to find the x value. y = 9 − x 2 0 = 9 − x 2 − y x 2 = 9 − y If y < 9 then 9 − y is positive. The equation y=9-x2 has two solutions for x, x = √ 9 − y or x = − √ 9 − y. For example, if y = 5 we find x = 2 or x = −2. The inputs 2 and −2 give the same result 5. This proves that the function y=9-x2 is a one to one function. Functions are foundational in science and mathematics. Whenever you have two sets of items that is x and y, then the functions demonstrate the relationships between them by giving the x values their corresponding y values. The various methods of verifying that functions are one to one functions show that you can never have two different input values of x that yield the same y results.

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