One to One Functions  Graph, Examples  Horizontal Line Test
What is a One to One Function?
A onetoone function is a mathematical function where each input corresponds to just one output. So, for each x, there is only 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 noted as the domain of the function, and the output value is noted as the range of the function.
Let's look at the pictures below:
For f(x), any value in the left circle corresponds to a unique value in the right circle. In conjunction, any value on the right side corresponds to a unique value on the left. In mathematical jargon, this implies every domain has a unique range, and every range holds a unique domain. Thus, this is an example of a onetoone function.
Here are some different examples of onetoone functions:

f(x) = x + 1

f(x) = 2x
Now let's study the second image, which displays the values for g(x).
Notice that the inputs in the left circle (domain) do not own unique outputs in the right circle (range). For example, the inputs 2 and 2 have equal output, in other words, 4. In conjunction, the inputs 4 and 4 have identical output, i.e., 16. We can see that there are identical Y values for numerous X values. Therefore, this is not a onetoone function.
Here are different 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 characteristics:

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 with respect to the line y = x.
How to Graph a One to One Function
In order to graph a onetoone function, you are required to find the domain and range for the function. Let's examine a straightforward example of a function f(x) = x + 1.
Immediately after you possess the domain and the range for the function, you need to chart the domain values on the Xaxis and range values on the Yaxis.
How can you tell whether a Function is One to One?
To indicate 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 passes through the graph of the function at more than one spot, then the function is not onetoone.
Since the graph of every linear function is a straight line, and a horizontal line doesn’t intersect the graph at more than one place, 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 look at the graph for f(x) = x + 1. Once you graph the values for the xcoordinates and ycoordinates, you need to consider whether or not a horizontal line intersects the graph at more than one spot. In this instance, the graph does not intersect any horizontal line more than once. This signifies that the function is a onetoone function.
On the other hand, if the function is not a onetoone function, it will intersect the same horizontal line more than one time. Let's examine the figure 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. Case in point, for either 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 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 also happens to be a onetoone function. The opposite of the function essentially undoes the function.
For example, in the case of f(x) = x + 1, we add 1 to each value of x in order to get the output, or y. The opposite 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 properties of an inverse onetoone function are the same as any other onetoone functions. This signifies that the opposite of a onetoone function will possess one domain for each range and pass the horizontal line test.
How do you find the inverse of a OnetoOne Function?
Determining the inverse of a function is not difficult. You just have to switch the x and y values. For instance, the inverse of the function f(x) = x + 5 is f1(x) = x  5.
Considering what we learned earlier, the inverse of a onetoone function undoes the function. Since the original output value required adding 5 to each input value, the new output value will require us to subtract 5 from each input value.
One to One Function Practice Questions
Contemplate 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. Plot the function and its inverse.
3. Find the inverse of the function numerically.
4. Specify the domain and range of both the function and its inverse.
5. Employ the inverse to determine the value for x in each equation.
Grade Potential Can Help You Master You Functions
If you happen to be facing difficulties using onetoone functions or similar topics, Grade Potential can set you up with a one on one teacher who can support you. Our Alpharetta math tutors are experienced professionals who assist students just like you advance their understanding of these types of functions.
With Grade Potential, you can learn at your own pace from the convenience of your own home. Book a call with Grade Potential today by calling (770) 9999794 to learn more about our tutoring services. One of our team members will get in touch with you to better inquire about your needs to set you up with the best tutor for you!