One to One Functions  Graph, Examples  Horizontal Line Test
What is a One to One Function?
A onetoone function is a mathematical function in which each input correlates to only one output. That is to say, for every x, there is just one y and vice versa. This signifies that the graph of a onetoone function will never intersect.
The input value in a onetoone function is known as the domain of the function, and the output value is known as the range of the function.
Let's examine the examples below:
For f(x), any value in the left circle corresponds to a unique value in the right circle. In conjunction, each value on the right side corresponds to a unique value on the left. In mathematical terms, this implies every domain owns a unique range, and every range has a unique domain. Therefore, this is an example of a onetoone function.
Here are some additional representations of onetoone functions:

f(x) = x + 1

f(x) = 2x
Now let's study the second image, which exhibits the values for g(x).
Notice that the inputs in the left circle (domain) do not have unique outputs in the right circle (range). For instance, the inputs 2 and 2 have the same output, i.e., 4. In the same manner, the inputs 4 and 4 have equal output, i.e., 16. We can discern that there are identical Y values for numerous X values. Hence, this is not a onetoone function.
Here are additional representations of non onetoone functions:

f(x) = x^2

f(x)=(x+2)^2
What are the characteristics of One to One Functions?
Onetoone functions have these qualities:

The function owns an inverse.

The graph of the function is a line that does not intersect itself.

The function passes the horizontal line test.

The graph of a function and its inverse are identical regarding the line y = x.
How to Graph a One to One Function
In order to graph a onetoone function, you will need to find the domain and range for the function. Let's study a simple example of a function f(x) = x + 1.
As soon as you know the domain and the range for the function, you need to plot the domain values on the Xaxis and range values on the Yaxis.
How can you evaluate whether a Function is One to One?
To test whether or not a function is onetoone, we can use the horizontal line test. As soon as you graph the graph of a function, draw horizontal lines over the graph. If a horizontal line moves through the graph of the function at more than one place, then the function is not onetoone.
Due to the fact that the graph of every linear function is a straight line, and a horizontal line does not intersect the graph at more than one spot, we can also conclude all linear functions are onetoone functions. Don’t forget that we do not apply the vertical line test for onetoone functions.
Let's look at the graph for f(x) = x + 1. As soon as you chart the values of xcoordinates and ycoordinates, you have to review if a horizontal line intersects the graph at more than one spot. In this example, the graph does not intersect any horizontal line more than once. This means that the function is a onetoone function.
On the contrary, if the function is not a onetoone function, it will intersect the same horizontal line more than once. Let's study the diagram for the f(y) = y^2. Here are the domain and the range values for the function:
Here is the graph for the function:
In this instance, the graph crosses numerous horizontal lines. For example, for both domains 1 and 1, the range is 1. Additionally, for both 2 and 2, the range is 4. This means that f(x) = x^2 is not a onetoone function.
What is the opposite of a OnetoOne Function?
As a onetoone function has just one input value for each output value, the inverse of a onetoone function also happens to be a onetoone function. The inverse of the function essentially reverses the function.
For Instance, in the case of f(x) = x + 1, we add 1 to each value of x for the purpose of getting the output, i.e., y. The inverse of this function will subtract 1 from each value of y.
The inverse of the function is known as f−1.
What are the properties of the inverse of a One to One Function?
The characteristics of an inverse onetoone function are the same as any other onetoone functions. This signifies that the inverse of a onetoone function will possess one domain for every range and pass the horizontal line test.
How do you find the inverse of a OnetoOne Function?
Determining the inverse of a function is very easy. You simply have to swap the x and y values. Case in point, the inverse of the function f(x) = x + 5 is f1(x) = x  5.
Considering what we learned previously, the inverse of a onetoone function reverses the function. Since the original output value required adding 5 to each input value, the new output value will require us to delete 5 from each input value.
One to One Function Practice Questions
Consider the following functions:

f(x) = x + 1

f(x) = 2x

f(x) = x2

f(x) = 3x  2

f(x) = x

g(x) = 2x + 1

h(x) = x/2  1

j(x) = √x

k(x) = (x + 2)/(x  2)

l(x) = 3√x

m(x) = 5  x
For each of these functions:
1. Determine whether the function is onetoone.
2. Chart the function and its inverse.
3. Find the inverse of the function algebraically.
4. State the domain and range of every function and its inverse.
5. Use the inverse to solve for x in each equation.
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