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. 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 noted as the domain of the function, and the output value is known as the range of the function.
Let's look at the images 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 correlates to a unique value in the left circle. In mathematical jargon, this means that every domain holds a unique range, and every range has a unique domain. Hence, this is an example of a onetoone function.
Here are some other examples of onetoone functions:

f(x) = x + 1

f(x) = 2x
Now let's examine the second image, which exhibits the values for g(x).
Pay attention to the fact that the inputs in the left circle (domain) do not have unique outputs in the right circle (range). Case in point, the inputs 2 and 2 have the same output, that is, 4. Similarly, the inputs 4 and 4 have the same output, i.e., 16. We can discern that there are equivalent Y values for many X values. Thus, this is not a onetoone function.
Here are different examples 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 properties:

The function holds 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 regarding 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 study a straightforward example of a function f(x) = x + 1.
Once you have the domain and the range for the function, you ought to graph the domain values on the Xaxis and range values on the Yaxis.
How can you tell whether a Function is One to One?
To prove if a function is onetoone, we can use the horizontal line test. Once you chart the graph of a function, trace horizontal lines over the graph. If a horizontal line moves through the graph of the function at more than one spot, 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 reason that all linear functions are onetoone functions. Remember that we do not apply the vertical line test for onetoone functions.
Let's examine the graph for f(x) = x + 1. Immediately after you chart the values for the xcoordinates and ycoordinates, you need to examine whether a horizontal line intersects the graph at more than one point. 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 other hand, if the function is not a onetoone function, it will intersect the same horizontal line multiple times. 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 intersects numerous horizontal lines. Case in point, for each domains 1 and 1, the range is 1. Additionally, for either 2 and 2, the range is 4. This signifies 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 reverses the function.
For example, in the example of f(x) = x + 1, we add 1 to each value of x in order to get the output, i.e., y. The opposite of this function will deduct 1 from each value of y.
The inverse of the function is denoted as f−1.
What are the properties of the inverse of a One to One Function?
The characteristics of an inverse onetoone function are identical to any other onetoone functions. This means that the opposite of a onetoone function will possess one domain for each range and pass the horizontal line test.
How do you figure out the inverse of a OnetoOne Function?
Finding the inverse of a function is simple. 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.
As we learned previously, the inverse of a onetoone function reverses the function. Since the original output value required us to add 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
Contemplate 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. Determine whether or not the function is onetoone.
2. Graph the function and its inverse.
3. Determine 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 formula.
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