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 whereby each input corresponds to only one output. That is to say, for each 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 the domain of the function, and the output value is the range of the function.
Let's examine the images below:
For f(x), every value in the left circle corresponds to a unique value in the right circle. In conjunction, each value on the right corresponds to a unique value on the left side. In mathematical words, this means that every domain holds a unique range, and every range holds 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 examine the second picture, which displays the values for g(x).
Pay attention to the fact that the inputs in the left circle (domain) do not have unique outputs in the right circle (range). For example, the inputs -2 and 2 have equal output, that is, 4. Similarly, 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. Hence, 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 characteristics 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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They pass 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
In order to graph a one-to-one function, you are required to figure out the domain and range for the function. Let's study a simple example of a function f(x) = x + 1.
Immediately after 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 determine whether or not a Function is One to One?
To indicate whether a function is one-to-one, we can apply the horizontal line test. Once you plot the graph of a function, draw horizontal lines over the graph. If a horizontal line intersects 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 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 apply the vertical line test for one-to-one functions.
Let's look at the graph for f(x) = x + 1. As soon as you chart the values for the x-coordinates and y-coordinates, you need to examine whether 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 means 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 more than one time. Let's study 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 example, the graph intersects various horizontal lines. For example, for each domains -1 and 1, the range is 1. In the same manner, for each -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?
As 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 undoes the function.
For example, in the event of f(x) = x + 1, we add 1 to each value of x as a means of getting the output, or y. The opposite of this function will deduct 1 from each value of y.
The inverse of the function is f−1.
What are the qualities 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 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 find the inverse of a One-to-One Function?
Figuring out the inverse of a function is simple. You just need to swap the x and y values. For instance, the inverse of the function f(x) = x + 5 is f-1(x) = x - 5.
Just like we reviewed previously, 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 deduct 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. Figure out whether the function is one-to-one.
2. Draw the function and its inverse.
3. Determine the inverse of the function mathematically.
4. Specify the domain and range of each function and its inverse.
5. Apply the inverse to solve for x in each calculation.
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