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 correlates to only one output. In other words, for each x, there is only one y and vice versa. This implies that the graph of a onetoone function will never intersect.
The input value in a onetoone function is the domain of the function, and the output value is the range of the function.
Let's look at the pictures below:
For f(x), every value in the left circle correlates to a unique value in the right circle. Similarly, any value on the right side corresponds to a unique value on the left. In mathematical jargon, this signifies every domain holds a unique range, and every range holds a unique domain. Hence, this is a representation of a onetoone function.
Here are some different examples of onetoone functions:

f(x) = x + 1

f(x) = 2x
Now let's look at the second image, which exhibits 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, that is, 4. Similarly, the inputs 4 and 4 have equal output, i.e., 16. We can comprehend that there are matching Y values for multiple X values. Hence, 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 characteristics of One to One Functions?
Onetoone functions have these properties:

The function has 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 identical concerning the line y = x.
How to Graph a One to One Function
When trying to graph a onetoone function, you are required to figure out the domain and range for the function. Let's look at a straightforward representation of a function f(x) = x + 1.
Once you possess the domain and the range for the function, you have to chart the domain values on the Xaxis and range values on the Yaxis.
How can you determine whether or not a Function is One to One?
To test if a function is onetoone, we can leverage the horizontal line test. Immediately after you plot the graph of a function, trace horizontal lines over the graph. If 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 doesn’t intersect the graph at more than one point, we can also conclude all linear functions are onetoone functions. Don’t forget that we do not use the vertical line test for onetoone functions.
Let's examine the graph for f(x) = x + 1. Once you graph the values of xcoordinates and ycoordinates, you ought to review whether 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 indicates 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 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 case, the graph meets multiple 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?
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 inverse of the function basically undoes the function.
Case in point, in the example of f(x) = x + 1, we add 1 to each value of x in order to get the output, in other words, 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 characteristics of the inverse of a One to One Function?
The qualities of an inverse onetoone function are no different than every other onetoone functions. This implies that the inverse of a onetoone function will hold one domain for every 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 very easy. You simply have to change the x and y values. Case in point, the inverse of the function f(x) = x + 5 is f1(x) = x  5.
As we learned earlier, the inverse of a onetoone function undoes the function. Considering 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 Examples
Examine 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 any of these functions:
1. Figure out whether the function is onetoone.
2. Graph the function and its inverse.
3. Find the inverse of the function algebraically.
4. Specify the domain and range of both the function and its inverse.
5. Use the inverse to determine the value for x in each equation.
Grade Potential Can Help You Master You Functions
If you happen to be struggling using onetoone functions or similar concepts, Grade Potential can put you in contact with a one on one teacher who can assist you. Our Alameda math tutors are experienced educators who support students just like you improve their mastery of these types of functions.
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