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 only one output. In other words, 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 noted as the domain of the function, and the output value is noted as the range of the function.
Let's examine the images below:
For f(x), each value in the left circle correlates to a unique value in the right circle. Similarly, each value on the right side corresponds to a unique value on the left. In mathematical words, this implies every domain holds a unique range, and every range has a unique domain. Thus, 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 look at the second picture, which displays 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). For example, the inputs 2 and 2 have equal output, that is, 4. In the same manner, the inputs 4 and 4 have equal output, i.e., 16. We can comprehend that there are matching Y values for many X values. Hence, 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 properties of One to One Functions?
Onetoone functions have the following characteristics:

The function has 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 the same 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 an easy example of a function f(x) = x + 1.
Once you know the domain and the range for the function, you ought to plot the domain values on the Xaxis and range values on the Yaxis.
How can you evaluate if a Function is One to One?
To test whether a function is onetoone, we can leverage the horizontal line test. Immediately after you chart 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.
Because the graph of every linear function is a straight line, and a horizontal line will not intersect the graph at more than one place, we can also conclude all linear functions are onetoone functions. Keep in mind that we do not apply the vertical line test for onetoone functions.
Let's examine the graph for f(x) = x + 1. Once you plot the values of xcoordinates and ycoordinates, you have to consider whether a horizontal line intersects the graph at more than one spot. In this case, 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 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 intersects various horizontal lines. For instance, for either domains 1 and 1, the range is 1. In the same manner, for either 2 and 2, the range is 4. This signifies that f(x) = x^2 is not a onetoone function.
What is the opposite of a OnetoOne Function?
Since a onetoone function has a single input value for each output value, the inverse of a onetoone function is also a onetoone function. The inverse of the function essentially reverses the function.
Case in point, in the event 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 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 every other onetoone functions. This means that the inverse of a onetoone function will possess one domain for every range and pass the horizontal line test.
How do you figure out 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.
As we learned earlier, the inverse of a onetoone function reverses the function. Because 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
Examine the subsequent 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. Figure out whether the function is onetoone.
2. Chart the function and its inverse.
3. Figure out the inverse of the function algebraically.
4. Indicate the domain and range of every function and its inverse.
5. Employ the inverse to determine the value for x in each formula.
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