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 correlates to a single output. That is to say, for every x, there is just one 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 the range of the function.
Let's examine the examples below:
For f(x), each value in the left circle correlates to a unique value in the right circle. In conjunction, any value on the right corresponds to a unique value on the left. In mathematical words, this means that every domain owns 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 examine the second image, which shows 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, in other words, 4. Similarly, the inputs -4 and 4 have the same output, i.e., 16. We can comprehend that there are equivalent Y values for numerous X values. Therefore, this is not a one-to-one function.
Here are different examples 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 properties of One to One Functions?
One-to-one functions have the following qualities:
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The function has 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 regarding the line y = x.
How to Graph a One to One Function
To graph a one-to-one function, you are required to determine the domain and range for the function. Let's examine an easy example of a function f(x) = x + 1.
Once you have the domain and the range for the function, you have to chart 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 indicate whether a function is one-to-one, we can leverage the horizontal line test. Once you plot 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 spot, then the function is not one-to-one.
Since the graph of every linear function is a straight line, and a horizontal line doesn’t intersect the graph at more than one place, 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. Immediately after you plot the values of x-coordinates and y-coordinates, you ought to consider if a horizontal line intersects the graph at more than one spot. In this instance, the graph does not intersect any horizontal line more than once. This means 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 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 meets numerous horizontal lines. For instance, for both domains -1 and 1, the range is 1. Similarly, 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 opposite 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 essentially undoes the function.
Case in point, in the example of f(x) = x + 1, we add 1 to each value of x for the purpose of getting the output, i.e., y. The inverse of this function will remove 1 from each value of y.
The inverse of the function is known as 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 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 find the inverse of a One-to-One Function?
Figuring out the inverse of a function is simple. You simply 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.
As we reviewed earlier, the inverse of a one-to-one function undoes 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 deduct 5 from each input value.
One to One Function Practice Examples
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 any of these functions:
1. Identify whether or not the function is one-to-one.
2. Chart the function and its inverse.
3. Find the inverse of the function algebraically.
4. State the domain and range of each function and its inverse.
5. Use the inverse to determine the value for x in each calculation.
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