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 a single output. In other words, for every 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 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. In conjunction, every value on the right correlates to a unique value on the left side. In mathematical words, this implies every domain has a unique range, and every range owns a unique domain. Therefore, 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 study the second picture, which shows the values for g(x).
Notice 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, that is, 4. In the same manner, the inputs 4 and 4 have the same output, i.e., 16. We can see that there are identical Y values for many X values. Therefore, 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 these characteristics:

The function holds an inverse.

The graph of the function is a line that does not intersect itself.

They pass 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 have to figure out the domain and range for the function. Let's examine an easy example of a function f(x) = x + 1.
As soon as 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 determine whether a Function is One to One?
To test whether or not 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. In the event that a horizontal line moves through the graph of the function at more than one point, then the function is not onetoone.
Because 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. 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. Once you chart the values to xcoordinates and ycoordinates, you need to review whether a horizontal line intersects the graph at more than one place. In this instance, the graph does not intersect any horizontal line more than once. This indicates 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 graph 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. For instance, for each domains 1 and 1, the range is 1. Similarly, for either 2 and 2, the range is 4. This means 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 also happens to be a onetoone function. The inverse of the function essentially undoes the function.
For Instance, in the case of f(x) = x + 1, we add 1 to each value of x as a means of getting 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 qualities of the inverse of a One to One Function?
The characteristics of an inverse onetoone function are identical to 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 find the inverse of a OnetoOne Function?
Finding the inverse of a function is simple. You simply have to switch the x and y values. For example, the inverse of the function f(x) = x + 5 is f1(x) = x  5.
As we learned before, the inverse of a onetoone function undoes the function. Since 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. Determine whether or not the function is onetoone.
2. Draw the function and its inverse.
3. Determine the inverse of the function numerically.
4. State the domain and range of both the function and its inverse.
5. Apply the inverse to find the solution for x in each formula.
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