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 corresponds to just one output. So, for every x, there is a single y and vice versa. This implies 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 known as the range of the function.
Let's study the images below:
For f(x), each value in the left circle corresponds to a unique value in the right circle. In the same manner, each value on the right side corresponds to a unique value in the left circle. In mathematical terms, this implies every domain has a unique range, and every range has a unique domain. Hence, this is an example 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 shows 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 instance, the inputs -2 and 2 have identical output, i.e., 4. Similarly, the inputs -4 and 4 have the same output, i.e., 16. We can comprehend that there are matching Y values for numerous X values. Thus, 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 these characteristics:
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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 the same with respect to the line y = x.
How to Graph a One to One Function
To graph a one-to-one function, you are required to determine the domain and range for the function. Let's look at an easy example of a function f(x) = x + 1.
Once you know the domain and the range for the function, you need to plot the domain values on the X-axis and range values on the Y-axis.
How can you evaluate if a Function is One to One?
To prove whether a function is one-to-one, we can use the horizontal line test. Immediately after you plot the graph of a function, draw horizontal lines over the graph. If a horizontal line moves 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 place, we can also reason that all linear functions are one-to-one functions. Keep in mind that we do not use the vertical line test for one-to-one functions.
Let's study the graph for f(x) = x + 1. Immediately after you plot the values to x-coordinates and y-coordinates, you have to examine 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 one time. 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 intersects numerous horizontal lines. Case in point, for both domains -1 and 1, the range is 1. Additionally, for both -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 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 opposite of the function basically undoes the function.
For example, in the example of f(x) = x + 1, we add 1 to each value of x in order to get 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 qualities of the inverse of a One to One Function?
The properties of an inverse one-to-one function are identical to all other one-to-one functions. This means that the reverse of a one-to-one function will hold one domain for every 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 very easy. You just need to switch the x and y values. For example, the inverse of the function f(x) = x + 5 is f-1(x) = x - 5.
Considering what we reviewed before, the inverse of a one-to-one function undoes 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
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. Determine whether the function is one-to-one.
2. Graph the function and its inverse.
3. Figure out the inverse of the function numerically.
4. State the domain and range of every function and its inverse.
5. Use the inverse to solve for x in each equation.
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