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 corresponds to a single output. In other words, for each x, there is just one y and vice versa. This signifies 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 study the images below:
For f(x), any value in the left circle corresponds to a unique value in the right circle. In the same manner, each value in the right circle correlates to a unique value on the left side. In mathematical words, this implies every domain owns a unique range, and every range holds a unique domain. Thus, this is an example of a one-to-one function.
Here are some different examples of one-to-one functions:
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f(x) = x + 1
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f(x) = 2x
Now let's look at the second picture, which shows the values for g(x).
Notice that the inputs in the left circle (domain) do not own unique outputs in the right circle (range). For example, the inputs -2 and 2 have identical 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 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 qualities of One to One Functions?
One-to-one functions have these characteristics:
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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
In order to graph a one-to-one function, you are required 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 have the domain and the range for the function, you need to plot 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 indicate if a function is one-to-one, we can use the horizontal line test. As soon as you plot the graph of a function, draw horizontal lines over the graph. In the event that a horizontal line passes through the graph of the function at more than one place, then the function is not one-to-one.
Because the graph of every linear function is a straight line, and a horizontal line will not 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 use the vertical line test for one-to-one functions.
Let's study the graph for f(x) = x + 1. As soon as you chart the values for the x-coordinates and y-coordinates, you ought to consider if 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.
Subsequently, if the function is not a one-to-one function, it will intersect the same horizontal line more than one time. Let's look at 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 case, the graph intersects various horizontal lines. For example, for either domains -1 and 1, the range is 1. Additionally, for either -2 and 2, the range is 4. This means that f(x) = x^2 is not a one-to-one function.
What is the inverse 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 reverses the function.
For Instance, in the case of f(x) = x + 1, we add 1 to each value of x in order to get the output, or 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 properties of an inverse one-to-one function are no different than any 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?
Finding the inverse of a function is very easy. You just have to swap the x and y values. For example, the inverse of the function f(x) = x + 5 is f-1(x) = x - 5.
Considering what we reviewed earlier, the inverse of a one-to-one function reverses the function. Because the original output value showed us we needed to add 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
Consider 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 if the function is one-to-one.
2. Chart the function and its inverse.
3. Figure out the inverse of the function mathematically.
4. State the domain and range of both the function and its inverse.
5. Employ the inverse to solve for x in each equation.
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