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 correlates to only one output. In other words, for each x, there is just one 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 known as the domain of the function, and the output value is the range of the function.
Let's study the pictures below:
For f(x), every value in the left circle correlates to a unique value in the right circle. In the same manner, every value in the right circle correlates to a unique value on the left. In mathematical terms, this implies every domain owns a unique range, and every range holds a unique domain. Thus, this is an example of a one-to-one function.
Here are some different examples 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 image, which displays the values for g(x).
Pay attention to the fact that the inputs in the left circle (domain) do not own unique outputs in the right circle (range). Case in point, the inputs -2 and 2 have equal output, in other words, 4. In conjunction, the inputs -4 and 4 have equal output, i.e., 16. We can see that there are equivalent Y values for many X values. Therefore, this is not a one-to-one function.
Here are additional representations 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 qualities:
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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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It passes 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
In order to graph a one-to-one function, you will need to figure out the domain and range for the function. Let's examine a straight-forward example of a function f(x) = x + 1.
Once you have the domain and the range for the function, you ought 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 indicate whether a function is one-to-one, we can apply the horizontal line test. Once you plot the graph of a function, trace horizontal lines over the graph. If a horizontal line intersects the graph of the function at more than one spot, then the function is not one-to-one.
Because the graph of every linear function is a straight line, and a horizontal line will not intersect the graph at more than one spot, we can also conclude all linear functions are one-to-one functions. Remember that we do not use the vertical line test for one-to-one functions.
Let's study the graph for f(x) = x + 1. As soon as you chart the values of x-coordinates and y-coordinates, you have to consider if a horizontal line intersects the graph at more than one spot. In this instance, 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 once. Let's look at the figure 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. Similarly, 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?
Since a one-to-one function has just 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 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 remove 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 qualities of an inverse one-to-one function are identical to any other one-to-one functions. This implies that the opposite 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?
Determining the inverse of a function is not difficult. You simply need 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 discussed before, the inverse of a one-to-one function reverses the function. Because the original output value required us to add 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
Contemplate 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. Determine whether or not the function is one-to-one.
2. Plot the function and its inverse.
3. Figure out the inverse of the function numerically.
4. Indicate the domain and range of both the function and its inverse.
5. Employ the inverse to determine the value for x in each equation.
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