Relationships one to one function refer to relationships between any two items in which one can only belong to one other item. In a mathematical sense, these relationships are known as one-to-one functions, in which there are equal numbers of items or in which one item is paired with only one other item. The name of a person and the reserved seat number of that person on a train is a simple daily life example of one function.
This article will help you understand what makes one-to-one functions unique and appreciate them. We will examine how to determine these functions based on expressions and graphs using solved examples.
What Is a One to One Function?
In fact, a normal function can have two different input values that can each produce the same answer, whereas a one-to-one function cannot. Let’s start by defining one-to-one functions and looking at their properties.
One to One Function Definition
One-to-one functions return a unique range for each element in their domain, i.e., the answer will never repeat. An example of a one-to-one function is g(x) = x – 4 since each input will result in a different answer. Also, the function g(x) = x2 is not a one-to-one function since it produces 4 as the answer when the inputs are 2 and -2. A function that is not a one-to-one is called a many to one function.
An algebraic definition of a one to one function is:
function g: D -> F is said to be one-to-one if
for all elements, x1x1 and x2x2 ∈ D. One-to-one functions are also considered injections, and a function is injective only if it is one-to-one. The contrapositive of this definition is a function g: D -> F is one-to-one if x1≠x2⇒g(x1)≠g(x2)x1≠x2⇒g(x1)≠g(x2). Let’s compare functions that are similar and those that are different by mapping two pairs of values.
For each fx value, there is only one unique value of f(x) and thus, f(x) is one to one function.
Different values of x, 2, and -2 are mapped with a common g(x) value 4 and the different x values -4 and 4 are mapped to a common value 16. Consequently, g(x) is not a one-to-one function.
Properties of One to One Function
A one-to-one function i.e an injective function that maps the distinct elements of its domain to the distinct elements of its co-domain. Below are some properties that help us to understand the various characteristics of one-to-one functions:
- If two functions, f(x) and k(x), are one to one, the f ◦ k is a one to one function as well. f ◦ k (x1x1) = f ◦ k (x2x2) ⇒ f(k(x1x1)) = f(k(x2x2)) ⇒ f(x1x1) = k(x2x2) ⇒ x1x1 = x2x2
- The domain of the function g is equal to the range of g-1, and the range of g equals the domain of g-1
- When a function is considered to be one to one, its graph will either be increasing or decreasing.
- g-1 (g(x)) = x, for every x in the domain of g, and g(g-1(x)) = x for every x in the domain of g-1
- If f ◦ k is a one to one function, then k(x) is also guaranteed to be a one to one function
- With respect to the line k = j, the graph of a function and the graph of its inverse are symmetric with one another
How to Determine if a Function Is One to One?
A vertical line test is used to determine whether a given relation is a function. In addition, we can determine if a function is one to one by using two methods:
- In terms of geometric logic, a one-to-one function would have a graph that passes through each value of y once every time.
- One-to-one functions are defined as functions where a = b for every g(a) = g(b)
One to One Function Graph
All functions can be represented as graphs. This function is represented by drawing a line on a plane as per the cartesian system. Domain and range are indicated horizontally along the x-axis, and vertically along the y-axis. If a function g is one to one function then no two points (x1x1,y1y1) and (x2x2,y2y2) have the same y-value. Thus, the graph of the equation y = g(x) cannot be cut more than once by a horizontal line. The one-to-one function is expected to have a unique value for each ‘a’. Because each ‘a’ will have a unique value for ‘y’, one-to-one functions will never have ordered pairs that share the same y-coordinate.
Inverse of One to One Function
Understanding one-to-one functions are crucial to understanding inverse functions and solving certain types of equations. An inverse function of a function g exists only if g equals one. The inverse of a one-to-one function g is denoted by g−1, where the ordered pairs of g-1 are obtained by interchanging the coordinates in each ordered pair of g. Thus, g becomes the domain of -1, and g-1 becomes the domain of -1.
Properties of the Inverse of One to One Function
Inverse functions undo what one to one functions did to a value in their domain in order to get back to the original y-value. Listed below are the properties of the inverse of one to one functions:
- F has an inverse function if and only if it is a one-to-one function; only one-to-one functions can have inverses.
- Since g and f are inverses of each other, they can both be considered one-to-one functions
- If a and b are inverses of each other if and only if (a ◦ b) (x) = x, x in the domain of b and (b ◦ a) (x) = x, x in the domain of a. Here a ◦ b is the composition function that has ‘a’ composed with ‘b’. ‘a’ and ‘b’ are called composite functions. In calculus, we use this notation ◦ denote that two functions are combined together to make a new function
- If a and b are inverses of each other then the domain of a is equal to the range of b and the range of a is equal to the domain of b.
- If a and b are inverses of each other, then their graphs will be reflected on the line y = x.
- If the point (c,d) is on the graph of f then point (d,c) is on the graph of f-1.
Steps to Find the Inverse of One to One Function
The step by step method to derive the inverse function g-1(x) for a one to one function g(x) is as follows:
- Set g(x) equal to y
- Switch the x with y since every (x,y) has a (y,x) partner
- Solve for y
- In the equation just found, rename y as g-1 (x).