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 only one output. In other words, for every x, there is a single 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 noted as the range of the function.
Let's study the pictures 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 on the right side corresponds to a unique value on the left side. In mathematical jargon, 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 additional examples 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 example, which exhibits the values for g(x).
Notice that the inputs in the left circle (domain) do not hold unique outputs in the right circle (range). Case in point, the inputs -2 and 2 have the same output, that is, 4. Similarly, the inputs -4 and 4 have identical output, i.e., 16. We can comprehend that there are matching Y values for multiple 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 properties of One to One Functions?
One-to-one functions have these properties:
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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 the same concerning the line y = x.
How to Graph a One to One Function
When trying to graph a one-to-one function, you will need to find the domain and range for the function. Let's examine a simple representation of a function f(x) = x + 1.
Once you know 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 determine whether a Function is One to One?
To indicate whether a function is one-to-one, we can leverage the horizontal line test. As soon as you chart the graph of a function, draw 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.
Since the graph of every linear function is a straight line, and a horizontal line doesn’t intersect the graph at more than one spot, we can also reason that 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. Immediately after you plot the values to x-coordinates and y-coordinates, you ought to review whether or not a horizontal line intersects the graph at more than one point. In this example, 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 more than once. Let's look at 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 intersects numerous horizontal lines. For example, for either domains -1 and 1, the range is 1. In the same manner, for both -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?
Since 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 essentially undoes the function.
For Instance, 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 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 qualities of the inverse of a One to One Function?
The characteristics of an inverse one-to-one function are identical to all other one-to-one functions. This signifies 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 find the inverse of a One-to-One Function?
Finding the inverse of a function is very easy. You just have to switch the x and y values. For instance, the inverse of the function f(x) = x + 5 is f-1(x) = x - 5.
As we reviewed previously, 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 subtract 5 from each input value.
One to One Function Practice Questions
Consider these 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. Plot the function and its inverse.
3. Find the inverse of the function mathematically.
4. Indicate the domain and range of every function and its inverse.
5. Apply the inverse to determine the value for x in each calculation.
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