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 corresponds to a single output. In other words, for each x, there is just 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 the range of the function.
Let's examine the images below:
For f(x), each value in the left circle correlates to a unique value in the right circle. Similarly, each value in the right circle corresponds to a unique value on the left side. In mathematical words, this implies every domain has a unique range, and every range has a unique domain. Therefore, this is a representation of a one-to-one function.
Here are some different 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 example, which displays the values for g(x).
Notice that the inputs in the left circle (domain) do not own unique outputs in the right circle (range). For instance, the inputs -2 and 2 have equal output, that is, 4. Similarly, the inputs -4 and 4 have identical output, i.e., 16. We can see that there are matching Y values for many 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 properties of One to One Functions?
One-to-one functions have the following properties:
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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 equivalent 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 need to figure out the domain and range for the function. Let's examine an easy 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 evaluate whether or not a Function is One to One?
To test whether or not a function is one-to-one, we can use the horizontal line test. Immediately after you graph 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 point, 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 does not 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 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 of x-coordinates and y-coordinates, you have to examine 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 indicates that the function is a one-to-one function.
Subsequently, 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 example, the graph crosses various horizontal lines. Case in point, for each domains -1 and 1, the range is 1. Additionally, 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?
Considering the fact that 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 essentially undoes the function.
For example, in the example of f(x) = x + 1, we add 1 to each value of x as a means of getting the output, or 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 characteristics of the inverse of a One to One Function?
The properties of an inverse one-to-one function are the same as 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 determine the inverse of a One-to-One Function?
Determining the inverse of a function is not difficult. You just 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 learned earlier, the inverse of a one-to-one function reverses the function. Because the original output value required adding 5 to each input value, the new output value will require us to deduct 5 from each input value.
One to One Function Practice Examples
Contemplate the following 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 each of these functions:
1. Determine if the function is one-to-one.
2. Plot the function and its inverse.
3. Determine the inverse of the function mathematically.
4. State the domain and range of both the function and its inverse.
5. Employ the inverse to solve for x in each calculation.
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