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. So, for every 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 noted as the domain of the function, and the output value is noted as the range of the function.
Let's examine the images below:
For f(x), every value in the left circle corresponds to a unique value in the right circle. In conjunction, every value on the right correlates to a unique value on the left side. In mathematical words, this implies every domain holds a unique range, and every range has a unique domain. Therefore, this is a representation 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 study the second picture, which shows the values for g(x).
Pay attention to the fact that the inputs in the left circle (domain) do not hold unique outputs in the right circle (range). For example, the inputs -2 and 2 have equal output, in other words, 4. In the same manner, the inputs -4 and 4 have the same output, i.e., 16. We can comprehend that there are matching Y values for multiple X values. Therefore, this is not a one-to-one function.
Here are some other 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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It passes the horizontal line test.
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The graph of a function and its inverse are identical with respect to 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 study a simple example of a function f(x) = x + 1.
Immediately after you possess 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 evaluate if a Function is One to One?
To test if a function is one-to-one, we can leverage the horizontal line test. Immediately after you graph 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 spot, we can also deduct 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 examine the graph for f(x) = x + 1. As soon as you plot the values of x-coordinates and y-coordinates, you need to review whether or not a horizontal line intersects the graph at more than one point. 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.
On the other hand, if the function is not a one-to-one function, it will intersect the same horizontal line multiple times. 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 example, the graph meets 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 means that f(x) = x^2 is not a one-to-one function.
What is the opposite of a One-to-One Function?
Since a one-to-one function has just 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.
For example, in the case 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 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 the same as every other one-to-one functions. This implies that the reverse 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 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.
As we discussed previously, the inverse of a one-to-one function undoes the function. Since 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 Examples
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 any of these functions:
1. Figure out whether the function is one-to-one.
2. Plot the function and its inverse.
3. Find the inverse of the function algebraically.
4. Indicate the domain and range of both the function and its inverse.
5. Employ the inverse to determine the value for x in each equation.
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