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 in which each input corresponds to just one output. That is to say, for each x, there is only one 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 examples below:
For f(x), each value in the left circle corresponds to a unique value in the right circle. In conjunction, any value on the right correlates to a unique value on the left. In mathematical terms, this signifies every domain holds a unique range, and every range holds a unique domain. Therefore, this is an example of a one-to-one function.
Here are some additional examples of one-to-one functions:
-
f(x) = x + 1
-
f(x) = 2x
Now let's look at the second image, which exhibits the values for g(x).
Notice that the inputs in the left circle (domain) do not own unique outputs in the right circle (range). For instance, the inputs -2 and 2 have equal output, i.e., 4. In the same manner, the inputs -4 and 4 have the same output, i.e., 16. We can comprehend that there are equivalent Y values for multiple X values. Hence, this is not a one-to-one function.
Here are some other representations of non one-to-one functions:
-
f(x) = x^2
-
f(x)=(x+2)^2
What are the qualities of One to One Functions?
One-to-one functions have the following 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
When trying to graph a one-to-one function, you are required to figure out the domain and range for the function. Let's examine an easy example of a function f(x) = x + 1.
Immediately after you have the domain and the range for the function, you need to plot the domain values on the X-axis and range values on the Y-axis.
How can you evaluate whether a Function is One to One?
To prove whether or not a function is one-to-one, we can apply the horizontal line test. Immediately after you chart the graph of a function, trace horizontal lines over the graph. If a horizontal line moves through the graph of the function at more than one point, then the function is not one-to-one.
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 one-to-one functions. Keep in mind that we do not apply the vertical line test for one-to-one functions.
Let's look at the graph for f(x) = x + 1. Once you chart the values for the x-coordinates and y-coordinates, you need to review 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 signifies 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 once. 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 instance, the graph crosses multiple horizontal lines. For example, for either domains -1 and 1, the range is 1. Similarly, for both -2 and 2, the range is 4. This signifies that f(x) = x^2 is not a one-to-one function.
What is the inverse of a One-to-One Function?
Since a one-to-one function has only one input value for each output value, the inverse of a one-to-one function also happens to be a one-to-one function. The inverse of the function essentially reverses the function.
For Instance, in the event 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 opposite of this function will remove 1 from each value of y.
The inverse of the function is known 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 identical to any other one-to-one functions. This means that the inverse of a one-to-one function will have one domain for each range and pass the horizontal line test.
How do you determine the inverse of a One-to-One Function?
Finding the inverse of a function is simple. You simply need to swap the x and y values. Case in point, the inverse of the function f(x) = x + 5 is f-1(x) = x - 5.
As we learned earlier, the inverse of a one-to-one function undoes the function. Because the original output value showed us we needed 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 Examples
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. Figure out if the function is one-to-one.
2. Graph the function and its inverse.
3. Figure out the inverse of the function algebraically.
4. Indicate 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 using one-to-one functions or similar concepts, Grade Potential can set you up with a one on one tutor who can assist you. Our Minneapolis math tutors are skilled professionals who help students just like you improve their skills of these types of functions.
With Grade Potential, you can study at your own pace from the comfort of your own home. Schedule a call with Grade Potential today by calling (612) 887-3325 to get informed about our tutoring services. One of our representatives will contact you to better determine your requirements to set you up with the best instructor for you!
