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 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 known as the domain of the function, and the output value is the range of the function.
Let's study the pictures below:
For f(x), every value in the left circle correlates to a unique value in the right circle. In the same manner, every value in the right circle correlates to a unique value on the left. In mathematical terms, this implies every domain owns 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).
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 equal output, in other words, 4. In conjunction, the inputs 4 and 4 have equal output, i.e., 16. We can see that there are equivalent Y values for many 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 properties 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
In order to graph a onetoone function, you will need to figure out the domain and range for the function. Let's examine a straightforward example of a function f(x) = x + 1.
Once you have the domain and the range for the function, you ought to chart the domain values on the Xaxis and range values on the Yaxis.
How can you tell if a Function is One to One?
To indicate whether a function is onetoone, we can apply the horizontal line test. Once 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 spot, 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 spot, we can also conclude all linear functions are onetoone functions. Remember that we do not use the vertical line test for onetoone functions.
Let's study the graph for f(x) = x + 1. As soon as you chart the values of xcoordinates and ycoordinates, you have to consider if 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 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 more than once. Let's look at 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 meets numerous horizontal lines. For instance, for each domains 1 and 1, the range is 1. Similarly, for each 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?
Since 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 basically undoes the function.
For example, in the case 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 remove 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 qualities of an inverse onetoone function are identical to any other onetoone functions. This implies that the opposite of a onetoone function will hold one domain for each 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 simply need 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 discussed before, the inverse of a onetoone function reverses the function. Because the original output value required us to add 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
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. Plot the function and its inverse.
3. Figure out the inverse of the function numerically.
4. Indicate the domain and range of both the function and its inverse.
5. Employ the inverse to determine the value for x in each equation.
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