One to One Functions - Graph, Examples | Horizontal Line Test
What is a One to One Function?
A one-to-one function is a mathematical function where each input corresponds 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 one-to-one function will never intersect.
The input value in a one-to-one function is noted as the domain of the function, and the output value is known as the range of the function.
Let's look at the images below:
For f(x), each value in the left circle correlates to a unique value in the right circle. Similarly, every value in the right circle correlates 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. Hence, this is a representation of a one-to-one function.
Here are some different representations of one-to-one functions:
-
f(x) = x + 1
-
f(x) = 2x
Now let's examine 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 have unique outputs in the right circle (range). For instance, the inputs -2 and 2 have identical output, i.e., 4. In conjunction, the inputs -4 and 4 have identical output, i.e., 16. We can see that there are matching Y values for numerous X values. Hence, this is not a one-to-one function.
Here are different examples of non one-to-one functions:
-
f(x) = x^2
-
f(x)=(x+2)^2
What are the properties of One to One Functions?
One-to-one functions have these qualities:
-
The function holds 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 regarding the line y = x.
How to Graph a One to One Function
In order to graph a one-to-one function, you will need to determine the domain and range for the function. Let's examine a simple example of a function f(x) = x + 1.
Once you possess the domain and the range for the function, you have to plot the domain values on the X-axis and range values on the Y-axis.
How can you tell if a Function is One to One?
To prove whether or not a function is one-to-one, we can use the horizontal line test. As soon as you graph the graph of a function, trace horizontal lines over the graph. In the event that a horizontal line moves through the graph of the function at more than one spot, then the function is not one-to-one.
Since 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 deduct all linear functions are one-to-one functions. Don’t forget that we do not leverage the vertical line test for one-to-one functions.
Let's examine the graph for f(x) = x + 1. Immediately after you plot the values for the x-coordinates and y-coordinates, you have to examine if a horizontal line intersects the graph at more than one spot. In this example, the graph does not intersect any horizontal line more than once. This means that the function is a one-to-one function.
On the contrary, if the function is not a one-to-one function, it will intersect the same horizontal line more than one time. Let's look at 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 example, the graph meets various horizontal lines. For example, for each domains -1 and 1, the range is 1. In the same manner, for both -2 and 2, the range is 4. This implies that f(x) = x^2 is not a one-to-one function.
What is the inverse of a One-to-One Function?
As a one-to-one function has only one input value for each output value, the inverse of a one-to-one function is also a one-to-one function. The inverse of the function basically reverses the function.
For example, in the example 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 remove 1 from each value of y.
The inverse of the function is denoted as f−1.
What are the characteristics of the inverse of a One to One Function?
The qualities of an inverse one-to-one function are the same as all other one-to-one functions. This means that the inverse of a one-to-one function will hold one domain for every range and pass the horizontal line test.
How do you figure out the inverse of a One-to-One Function?
Figuring out the inverse of a function is not difficult. You just need to switch the x and y values. Case in point, the inverse of the function f(x) = x + 5 is f-1(x) = x - 5.
Just like we discussed previously, the inverse of a one-to-one function reverses the function. Considering the original output value showed us we needed to add 5 to each input value, the new output value will require us to subtract 5 from each input value.
One to One Function Practice Examples
Examine the following 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 every function:
1. Determine whether or not the function is one-to-one.
2. Draw the function and its inverse.
3. Determine the inverse of the function numerically.
4. Specify the domain and range of each function and its inverse.
5. Use the inverse to find the solution for x in each calculation.
Grade Potential Can Help You Master You Functions
If you find yourself having problems trying to understand one-to-one functions or similar topics, Grade Potential can connect you with a one on one teacher who can support you. Our San Bernardino math tutors are experienced educators who help students just like you improve their understanding of these types of functions.
With Grade Potential, you can learn at your own pace from the convenience of your own home. Schedule a meeting with Grade Potential today by calling (909) 316-8940 to get informed about our tutoring services. One of our team members will contact you to better inquire about your requirements to set you up with the best teacher for you!
