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 correlates to a single output. In other words, for each x, there is only one 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 known 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. In the same manner, any value on the right correlates to a unique value on the left. In mathematical words, this implies every domain holds a unique range, and every range owns a unique domain. Hence, this is a representation of a one-to-one function.
Here are some other 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 hold unique outputs in the right circle (range). For instance, the inputs -2 and 2 have the same output, i.e., 4. In conjunction, the inputs -4 and 4 have identical output, i.e., 16. We can discern that there are matching Y values for numerous X values. Hence, this is not a one-to-one function.
Here are some other 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 characteristics of One to One Functions?
One-to-one functions have these 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 with respect to 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 study a simple example of a function f(x) = x + 1.
As soon as you know 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 whether a Function is One to One?
To test whether or not a function is one-to-one, we can apply the horizontal line test. Once you graph the graph of a function, trace horizontal lines over the graph. In the event that a horizontal line intersects the graph of the function at more than one spot, 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 spot, we can also deduct 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 look at the graph for f(x) = x + 1. Immediately after you chart the values for the x-coordinates and y-coordinates, you ought to review whether a horizontal line intersects the graph at more than one point. In this instance, the graph does not intersect any horizontal line more than once. This signifies 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 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 numerous horizontal lines. Case in point, 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 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 inverse of the function basically reverses the function.
For Instance, in the example of f(x) = x + 1, we add 1 to each value of x for the purpose of getting the output, or y. The inverse 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 properties of an inverse one-to-one function are the same as all other one-to-one functions. This means that the inverse of a one-to-one function will have 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 very easy. You just need 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 reviewed before, the inverse of a one-to-one function reverses the function. Since 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 Examples
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 every function:
1. Identify whether or not the function is one-to-one.
2. Chart the function and its inverse.
3. Determine the inverse of the function mathematically.
4. Specify the domain and range of both the function and its inverse.
5. Employ the inverse to determine the value for x in each formula.
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