One to One Functions  Graph, Examples  Horizontal Line Test
What is a One to One Function?
A onetoone function is a mathematical function whereby each input correlates to just one output. So, for every x, there is only one y and vice versa. This means that the graph of a onetoone function will never intersect.
The input value in a onetoone function is noted as the domain of the function, and the output value is the range of the function.
Let's examine the pictures below:
For f(x), every value in the left circle corresponds to a unique value in the right circle. Similarly, each value on the right correlates to a unique value on the left. In mathematical words, this signifies every domain has a unique range, and every range has a unique domain. Hence, 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 examine the second picture, which shows the values for g(x).
Pay attention to the fact that the inputs in the left circle (domain) do not own unique outputs in the right circle (range). Case in point, the inputs 2 and 2 have identical output, i.e., 4. In conjunction, the inputs 4 and 4 have identical output, i.e., 16. We can see that there are identical Y values for multiple X values. Therefore, 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 qualities 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.

It passes the horizontal line test.

The graph of a function and its inverse are equivalent regarding the line y = x.
How to Graph a One to One Function
When trying to graph a onetoone function, you are required to find the domain and range for the function. Let's look at a simple representation of a function f(x) = x + 1.
Immediately after you possess the domain and the range for the function, you need to graph the domain values on the Xaxis and range values on the Yaxis.
How can you determine whether a Function is One to One?
To test whether or not a function is onetoone, we can leverage the horizontal line test. Once you plot the graph of a function, trace horizontal lines over the graph. In the event that a horizontal line intersects the graph of the function at more than one place, then the function is not onetoone.
Because the graph of every linear function is a straight line, and a horizontal line will not intersect the graph at more than one point, we can also conclude all linear functions are onetoone functions. Remember that we do not apply the vertical line test for onetoone functions.
Let's study the graph for f(x) = x + 1. Once you chart the values for the xcoordinates and ycoordinates, you have to review whether or not a horizontal line intersects the graph at more than one place. In this example, the graph does not intersect any horizontal line more than once. This signifies that the function is a onetoone function.
Subsequently, if the function is not a onetoone function, it will intersect the same horizontal line multiple times. Let's examine the graph 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 case, the graph meets numerous horizontal lines. For instance, for either domains 1 and 1, the range is 1. Similarly, for both 2 and 2, the range is 4. This implies that f(x) = x^2 is not a onetoone function.
What is the inverse of a OnetoOne Function?
Considering the fact that a onetoone function has just one input value for each output value, the inverse of a onetoone function is also a onetoone function. The inverse of the function basically undoes the function.
For Instance, in the case of f(x) = x + 1, we add 1 to each value of x as a means of getting the output, or y. The inverse of this function will subtract 1 from each value of y.
The inverse of the function is denoted as f−1.
What are the characteristics of the inverse of a One to One Function?
The characteristics of an inverse onetoone function are identical to all other onetoone functions. This means that the inverse of a onetoone function will hold one domain for each range and pass the horizontal line test.
How do you determine the inverse of a OnetoOne Function?
Figuring out the inverse of a function is simple. You simply have to change the x and y values. For example, the inverse of the function f(x) = x + 5 is f1(x) = x  5.
As we discussed earlier, the inverse of a onetoone function undoes the function. Since the original output value required us to add 5 to each input value, the new output value will require us to deduct 5 from each input value.
One to One Function Practice Examples
Consider these 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 any of these functions:
1. Figure out whether the function is onetoone.
2. Draw the function and its inverse.
3. Figure out the inverse of the function numerically.
4. Specify the domain and range of every function and its inverse.
5. Apply the inverse to find the solution for x in each equation.
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