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 just one output. So, for every x, there is only 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 known as the domain of the function, and the output value is the range of the function.
Let's examine the images below:
For f(x), any value in the left circle correlates to a unique value in the right circle. In conjunction, each value in the right circle corresponds to a unique value on the left. In mathematical words, this implies every domain holds a unique range, and every range holds a unique domain. Therefore, this is an example of a one-to-one function.
Here are some other 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 image, which exhibits 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 identical output, that is, 4. In conjunction, the inputs -4 and 4 have identical output, i.e., 16. We can comprehend that there are identical 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 these properties:
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The function owns an inverse.
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The graph of the function is a line that does not intersect itself.
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They pass 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
In order to graph a one-to-one function, you are required to determine the domain and range for the function. Let's look at a straight-forward representation of a function f(x) = x + 1.
As soon as you possess the domain and the range for the function, you ought to graph 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, draw horizontal lines over the graph. In the event that 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 will not 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 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 for the x-coordinates and y-coordinates, you have to examine 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 multiple times. Let's examine 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 numerous horizontal lines. For instance, for each domains -1 and 1, the range is 1. In the same manner, 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 inverse of a One-to-One Function?
Considering the fact that 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 basically reverses the function.
For example, in the case of f(x) = x + 1, we add 1 to each value of x in order to get the output, i.e., 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 properties of an inverse one-to-one function are the same as any other one-to-one functions. This signifies that the inverse of a one-to-one function will possess one domain for every range and pass the horizontal line test.
How do you find the inverse of a One-to-One Function?
Finding the inverse of a function is not difficult. You simply 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.
As we learned before, the inverse of a one-to-one function undoes the function. Considering the original output value required adding 5 to each input value, the new output value will require us to deduct 5 from each input value.
One to One Function Practice Questions
Examine 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 if the function is one-to-one.
2. Graph 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. Apply the inverse to find the solution for x in each calculation.
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