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. So, 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 noted as the range of the function.
Let's look at the pictures below:
For f(x), each value in the left circle correlates to a unique value in the right circle. Similarly, any value in the right circle corresponds to a unique value on the left. In mathematical terms, this signifies every domain holds a unique range, and every range has a unique domain. Thus, this is a representation of a onetoone function.
Here are some other representations of onetoone functions:

f(x) = x + 1

f(x) = 2x
Now let's examine the second image, which shows 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). For example, the inputs 2 and 2 have the same output, in other words, 4. Similarly, the inputs 4 and 4 have the same output, i.e., 16. We can comprehend that there are equivalent Y values for many X values. Thus, this is not a onetoone function.
Here are some other 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 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
In order to graph a onetoone function, you will need to find the domain and range for the function. Let's examine an easy example of a function f(x) = x + 1.
Once you have the domain and the range for the function, you have to graph the domain values on the Xaxis and range values on the Yaxis.
How can you tell if a Function is One to One?
To test whether or not a function is onetoone, we can apply the horizontal line test. Once you plot 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 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 doesn’t intersect the graph at more than one place, 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. Immediately after you plot the values for the xcoordinates and ycoordinates, you have to examine whether or not a horizontal line intersects the graph at more than one point. In this case, the graph does not intersect any horizontal line more than once. This indicates 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 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 example, the graph intersects multiple horizontal lines. For instance, for each 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 inverse of a OnetoOne Function?
Considering the fact that 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 basically undoes the function.
For Instance, in the event of f(x) = x + 1, we add 1 to each value of x in order to get the output, or 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 identical to every other onetoone functions. This implies that the reverse of a onetoone function will have one domain for each range and pass the horizontal line test.
How do you find the inverse of a OnetoOne Function?
Finding the inverse of a function is not difficult. You simply need to change the x and y values. Case in point, the inverse of the function f(x) = x + 5 is f1(x) = x  5.
Considering what we reviewed earlier, 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 deduct 5 from each input value.
One to One Function Practice Examples
Contemplate 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 any of these functions:
1. Determine whether the function is onetoone.
2. Plot the function and its inverse.
3. Determine the inverse of the function algebraically.
4. Indicate the domain and range of every function and its inverse.
5. Use the inverse to solve for x in each formula.
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