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 corresponds to just one output. In other words, for each x, there is a single 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 examine the examples below:
For f(x), any value in the left circle corresponds to a unique value in the right circle. In conjunction, any value on the right side correlates to a unique value on the left. In mathematical jargon, this signifies every domain has 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 representations 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 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). Case in point, the inputs -2 and 2 have the same output, i.e., 4. Similarly, the inputs -4 and 4 have equal output, i.e., 16. We can see that there are matching Y values for numerous X values. Hence, this is not a one-to-one function.
Here are additional 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 qualities of One to One Functions?
One-to-one functions have the following 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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They pass 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
When trying to graph a one-to-one function, you will have to determine the domain and range for the function. Let's examine a straight-forward example of a function f(x) = x + 1.
Immediately after you have the domain and the range for the function, you need 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 test whether or not a function is one-to-one, we can use the horizontal line test. Immediately after 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 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 conclude 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 look at the graph for f(x) = x + 1. As soon as you graph the values for the x-coordinates and y-coordinates, you ought to examine whether 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.
On the contrary, 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 intersects various horizontal lines. Case in point, for either domains -1 and 1, the range is 1. In the same manner, for both -2 and 2, the range is 4. This implies that f(x) = x^2 is not a one-to-one function.
What is the inverse of a One-to-One Function?
As a one-to-one function has a single 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 essentially reverses the function.
For Instance, 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 remove 1 from each value of y.
The inverse of the function is known as f−1.
What are the characteristics of the inverse of a One to One Function?
The properties of an inverse one-to-one function are identical to all other one-to-one functions. This means that the inverse of a one-to-one function will hold one domain for every range and pass the horizontal line test.
How do you find the inverse of a One-to-One Function?
Determining the inverse of a function is not difficult. You just have to change the x and y values. Case in point, the inverse of the function f(x) = x + 5 is f-1(x) = x - 5.
Considering what we discussed before, the inverse of a one-to-one function reverses the function. Since the original output value showed us we needed to add 5 to each input value, the new output value will require us to subtract 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. Chart the function and its inverse.
3. Find the inverse of the function algebraically.
4. State the domain and range of every function and its inverse.
5. Employ the inverse to solve for x in each calculation.
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