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. In other words, for each x, there is just one y and vice versa. This implies 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 noted as the range of the function.
Let's look at the pictures below:
For f(x), every value in the left circle correlates to a unique value in the right circle. Similarly, any value on the right side corresponds to a unique value on the left side. In mathematical jargon, this means that 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 other representations 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).
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, that is, 4. Similarly, the inputs -4 and 4 have identical output, i.e., 16. We can see that there are equivalent Y values for multiple X values. Thus, 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 qualities of One to One Functions?
One-to-one functions have the following characteristics:
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The function holds 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 concerning the line y = x.
How to Graph a One to One Function
To graph a one-to-one function, you will have to find the domain and range for the function. Let's study a simple representation of a function f(x) = x + 1.
Once you know the domain and the range for the function, you ought to plot the domain values on the X-axis and range values on the Y-axis.
How can you evaluate whether a Function is One to One?
To test whether or not a function is one-to-one, we can leverage the horizontal line test. Immediately after you chart 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.
Since 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 leverage the vertical line test for one-to-one functions.
Let's study the graph for f(x) = x + 1. As soon as you plot the values to x-coordinates and y-coordinates, you ought to review if a horizontal line intersects the graph at more than one place. In this example, the graph does not intersect any horizontal line more than once. This means 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 one time. 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 case, the graph crosses multiple horizontal lines. For instance, for both domains -1 and 1, the range is 1. Similarly, 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 is also a one-to-one function. The opposite of the function basically undoes the function.
Case in point, in the case of f(x) = x + 1, we add 1 to each value of x for the purpose of getting the output, i.e., y. The inverse of this function will subtract 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 qualities of an inverse one-to-one function are no different than every other one-to-one functions. This means that the reverse of a one-to-one function will hold one domain for each range and pass the horizontal line test.
How do you figure out the inverse of a One-to-One Function?
Finding the inverse of a function is not difficult. 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 before, the inverse of a one-to-one function reverses 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 Questions
Examine 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 every function:
1. Figure out if the function is one-to-one.
2. Plot the function and its inverse.
3. Figure out the inverse of the function numerically.
4. State the domain and range of both the function and its inverse.
5. Employ the inverse to solve for x in each formula.
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