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 whereby each input corresponds to only one output. So, for every x, there is only one y and vice versa. This means 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 study the examples below:
For f(x), any value in the left circle correlates to a unique value in the right circle. Similarly, every value in the right circle corresponds to a unique value in the left circle. In mathematical words, this signifies every domain has a unique range, and every range holds a unique domain. Therefore, this is a representation 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 study the second picture, which displays the values for g(x).
Pay attention to the fact 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, that is, 4. Similarly, the inputs -4 and 4 have the same output, i.e., 16. We can discern that there are identical Y values for multiple X values. Therefore, 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 properties of One to One Functions?
One-to-one functions have the following characteristics:
-
The function owns 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 the same regarding the line y = x.
How to Graph a One to One Function
When trying to graph a one-to-one function, you will need to find the domain and range for the function. Let's study a straight-forward representation of a function f(x) = x + 1.
Once you possess the domain and the range for the function, you ought to graph the domain values on the X-axis and range values on the Y-axis.
How can you determine whether a Function is One to One?
To prove whether a function is one-to-one, we can use the horizontal line test. As soon as you plot the graph of a function, trace horizontal lines over the graph. If a horizontal line intersects the graph of the function at more than one spot, 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 place, we can also deduct all linear functions are one-to-one functions. Remember 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 chart the values for the x-coordinates and y-coordinates, you have to consider whether a horizontal line intersects the graph at more than one place. In this case, 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 one time. Let's examine the figure 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 intersects various horizontal lines. For instance, for either domains -1 and 1, the range is 1. Additionally, for either -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?
As a one-to-one function has a single input value for each output value, the inverse of a one-to-one function also happens to be a one-to-one function. The opposite of the function basically reverses the function.
For example, in the event of f(x) = x + 1, we add 1 to each value of x for the purpose of getting the output, or y. The inverse of this function will remove 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 properties of an inverse one-to-one function are no different than all other one-to-one functions. This signifies that the opposite 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?
Determining the inverse of a function is not difficult. You simply need to switch the x and y values. For example, the inverse of the function f(x) = x + 5 is f-1(x) = x - 5.
As we learned before, the inverse of a one-to-one function reverses the function. Because the original output value required adding 5 to each input value, the new output value will require us to subtract 5 from each input value.
One to One Function Practice Questions
Consider 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 both the function and its inverse.
5. Use the inverse to solve for x in each equation.
Grade Potential Can Help You Learn You Functions
If you are facing difficulties trying to understand one-to-one functions or similar concepts, Grade Potential can connect you with a 1:1 teacher who can help. Our Dayton math tutors are skilled educators who support students just like you advance their mastery of these subjects.
With Grade Potential, you can learn at your unique pace from the comfort of your own home. Schedule an appointment with Grade Potential today by calling (937) 777-9494 to get informed about our tutoring services. One of our representatives will get in touch with you to better determine your needs to provide you with the best instructor for you!
