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 correlates to just one output. In other words, for every x, there is a single y and vice versa. This implies 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 look at the pictures below:
For f(x), every value in the left circle correlates to a unique value in the right circle. Similarly, every value on the right correlates to a unique value in the left circle. In mathematical jargon, this means that every domain has a unique range, and every range owns a unique domain. Hence, this is an example of a one-to-one function.
Here are some more representations of one-to-one functions:
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f(x) = x + 1
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f(x) = 2x
Now let's study the second image, which exhibits the values for g(x).
Be aware of the fact that the inputs in the left circle (domain) do not hold unique outputs in the right circle (range). For instance, the inputs -2 and 2 have identical output, i.e., 4. In the same manner, the inputs -4 and 4 have identical output, i.e., 16. We can comprehend that there are matching 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:
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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 concerning the line y = x.
How to Graph a One to One Function
To graph a one-to-one function, you will need to figure out the domain and range for the function. Let's examine an easy example of a function f(x) = x + 1.
Once you know the domain and the range for the function, you ought to chart 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 indicate whether or not a function is one-to-one, we can leverage the horizontal line test. As soon as you chart 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 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 doesn’t intersect the graph at more than one point, we can also reason that all linear functions are one-to-one functions. Remember that we do not use 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 to x-coordinates and y-coordinates, you have to consider whether or not 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 one-to-one function.
Subsequently, if the function is not a one-to-one function, it will intersect the same horizontal line more than once. 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 example, the graph crosses numerous horizontal lines. For instance, for either domains -1 and 1, the range is 1. Additionally, for each -2 and 2, the range is 4. This signifies that f(x) = x^2 is not a one-to-one function.
What is the opposite of a One-to-One Function?
Considering the fact that 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 opposite of the function basically undoes the function.
For example, in the example of f(x) = x + 1, we add 1 to each value of x in order to get the output, or y. The opposite of this function will deduct 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 qualities of an inverse one-to-one function are no different than every other one-to-one functions. This implies that the inverse of a one-to-one function will possess one domain for each range and pass the horizontal line test.
How do you determine the inverse of a One-to-One Function?
Determining the inverse of a function is simple. You just 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 previously, 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 delete 5 from each input value.
One to One Function Practice Questions
Contemplate the subsequent 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 each of these functions:
1. Determine whether the function is one-to-one.
2. Draw the function and its inverse.
3. Find the inverse of the function numerically.
4. State the domain and range of both the function and its inverse.
5. Use the inverse to determine the value for x in each calculation.
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