One to One Functions  Graph, Examples  Horizontal Line Test
What is a One to One Function?
A onetoone function is a mathematical function in which each input correlates to a single output. That is to say, for each x, there is a single 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 study the images below:
For f(x), any value in the left circle correlates to a unique value in the right circle. In the same manner, any value in the right circle corresponds to a unique value on the left. In mathematical terms, this signifies every domain owns a unique range, and every range holds a unique domain. Hence, this is an example of a onetoone function.
Here are some more representations of onetoone functions:

f(x) = x + 1

f(x) = 2x
Now let's look at the second example, which displays the values for g(x).
Be aware of the fact that the inputs in the left circle (domain) do not hold unique outputs in the right circle (range). Case in point, the inputs 2 and 2 have identical output, i.e., 4. Similarly, the inputs 4 and 4 have equal output, i.e., 16. We can discern that there are identical 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 qualities of One to One Functions?
Onetoone functions have the following qualities:

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 with respect to the line y = x.
How to Graph a One to One Function
In order to graph a onetoone function, you will have to find the domain and range for the function. Let's look at an easy example of a function f(x) = x + 1.
Once you have 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 whether or not a Function is One to One?
To prove whether or not a function is onetoone, we can use the horizontal line test. Immediately after you graph the graph of a function, trace horizontal lines over the graph. If a horizontal line passes 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 does not intersect the graph at more than one spot, we can also deduct all linear functions are onetoone functions. Keep in mind that we do not apply the vertical line test for onetoone functions.
Let's look at the graph for f(x) = x + 1. As soon as you chart the values of xcoordinates and ycoordinates, you have to review whether or not 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.
On the contrary, if the function is not a onetoone function, it will intersect the same horizontal line more than one time. Let's look at 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 intersects multiple horizontal lines. Case in point, for both domains 1 and 1, the range is 1. Additionally, for either 2 and 2, the range is 4. This implies that f(x) = x^2 is not a onetoone function.
What is the opposite of a OnetoOne Function?
As a onetoone function has just one input value for each output value, the inverse of a onetoone function is also a onetoone function. The opposite of the function basically reverses the function.
Case in point, in the case of f(x) = x + 1, we add 1 to each value of x for the purpose of getting the output, in other words, y. The inverse of this function will deduct 1 from each value of y.
The inverse of the function is f−1.
What are the qualities of the inverse of a One to One Function?
The qualities of an inverse onetoone function are the same as any other onetoone functions. This signifies that the reverse of a onetoone function will possess 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 simple. You just need to swap the x and y values. Case in point, the inverse of the function f(x) = x + 5 is f1(x) = x  5.
Just like we learned previously, the inverse of a onetoone function reverses the function. Considering 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. Identify whether the function is onetoone.
2. Draw the function and its inverse.
3. Find the inverse of the function numerically.
4. State the domain and range of each function and its inverse.
5. Employ the inverse to find the solution for x in each equation.
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