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. So, for every x, there is only 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 the domain of the function, and the output value is known as the range of the function.
Let's examine the examples below:
For f(x), every value in the left circle correlates to a unique value in the right circle. In conjunction, every value on the right corresponds to a unique value on the left side. In mathematical words, this signifies every domain owns a unique range, and every range holds a unique domain. Therefore, this is an example of a one-to-one function.
Here are some additional representations of one-to-one functions:
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f(x) = x + 1
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f(x) = 2x
Now let's look at the second picture, which shows 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 example, the inputs -2 and 2 have identical output, that is, 4. In conjunction, the inputs -4 and 4 have the same 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 different 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 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 equivalent regarding the line y = x.
How to Graph a One to One Function
When trying to graph a one-to-one function, you are required to find the domain and range for the function. Let's examine 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 plot 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 test whether or not a function is one-to-one, we can leverage the horizontal line test. Immediately after you plot the graph of a function, draw horizontal lines over the graph. If a horizontal line passes through the graph of the function at more than one spot, 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 deduct 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. Once you plot the values for the x-coordinates and y-coordinates, you have to examine if 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 signifies 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 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 case, the graph crosses multiple horizontal lines. For instance, for each domains -1 and 1, the range is 1. In the same manner, for both -2 and 2, the range is 4. This implies that f(x) = x^2 is not a one-to-one function.
What is the opposite 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 inverse of the function essentially undoes 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 opposite 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 characteristics of an inverse one-to-one function are no different than any 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?
Determining the inverse of a function is not difficult. You just have to swap 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. Since the original output value required adding 5 to each input value, the new output value will require us to subtract 5 from each input value.
One to One Function Practice Examples
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 any of these functions:
1. Identify if the function is one-to-one.
2. Chart the function and its inverse.
3. Find the inverse of the function mathematically.
4. Specify the domain and range of both the function and its inverse.
5. Use the inverse to find the solution for x in each formula.
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