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 correlates to only one output. In other words, for each x, there is just 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 the domain of the function, and the output value is noted as the range of the function.
Let's study the pictures below:
For f(x), each value in the left circle corresponds to a unique value in the right circle. In conjunction, any value in the right circle corresponds to a unique value on the left side. In mathematical words, this implies every domain holds a unique range, and every range owns a unique domain. Therefore, this is an example of a one-to-one function.
Here are some more examples of one-to-one functions:
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f(x) = x + 1
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f(x) = 2x
Now let's examine the second image, which displays the values for g(x).
Notice 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 the same output, i.e., 4. In the same manner, the inputs -4 and 4 have the same output, i.e., 16. We can see that there are equivalent Y values for numerous X values. Thus, this is not a one-to-one function.
Here are additional representations of non one-to-one functions:
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f(x) = x^2
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f(x)=(x+2)^2
What are the characteristics of One to One Functions?
One-to-one functions have the following properties:
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The function holds an inverse.
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The graph of the function is a line that does not intersect itself.
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The function passes the horizontal line test.
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The graph of a function and its inverse are equivalent with respect to 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 study a simple example of a function f(x) = x + 1.
Immediately after you have 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 determine whether or not a Function is One to One?
To prove whether a function is one-to-one, we can apply the horizontal line test. As soon as you chart the graph of a function, trace horizontal lines over the graph. In the event that a horizontal line passes through the graph of the function at more than one point, then the function is not one-to-one.
Due to the fact that the graph of every linear function is a straight line, and a horizontal line does not intersect the graph at more than one point, we can also deduct all linear functions are one-to-one functions. Keep in mind that we do not leverage the vertical line test for one-to-one functions.
Let's examine the graph for f(x) = x + 1. As soon as you graph the values of x-coordinates and y-coordinates, you need to consider whether a horizontal line intersects the graph at more than one place. In this example, the graph does not intersect any horizontal line more than once. This indicates that the function is a one-to-one function.
On the other hand, if the function is not a one-to-one function, it will intersect the same horizontal line more than one time. Let's study 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 instance, the graph meets various horizontal lines. For instance, for either domains -1 and 1, the range is 1. Similarly, 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 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 inverse of the function basically reverses the function.
Case in point, 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 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 properties of the inverse of a One to One Function?
The characteristics of an inverse one-to-one function are the same as every other one-to-one functions. This implies 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 simple. You simply have to change the x and y values. For example, the inverse of the function f(x) = x + 5 is f-1(x) = x - 5.
Just like we learned 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 Questions
Contemplate the following functions:
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f(x) = x + 1
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f(x) = 2x
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f(x) = x2
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f(x) = 3x - 2
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f(x) = |x|
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g(x) = 2x + 1
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h(x) = x/2 - 1
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j(x) = √x
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k(x) = (x + 2)/(x - 2)
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l(x) = 3√x
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m(x) = 5 - x
For every function:
1. Identify whether or not the function is one-to-one.
2. Plot the function and its inverse.
3. Figure out the inverse of the function numerically.
4. Specify the domain and range of both the function and its inverse.
5. Employ the inverse to find the solution for x in each formula.
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