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 just one output. So, for each x, there is a single 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 noted as the domain of the function, and the output value is noted as the range of the function.
Let's look at the examples below:
For f(x), every value in the left circle correlates to a unique value in the right circle. Similarly, each value on the right corresponds to a unique value in the left circle. In mathematical jargon, this means that every domain owns a unique range, and every range owns a unique domain. Thus, 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 examine the second image, which shows the values for g(x).
Notice 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 the same output, that is, 4. In the same manner, the inputs -4 and 4 have identical output, i.e., 16. We can comprehend that there are equivalent Y values for multiple X values. Thus, 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 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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The function 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 have to figure out the domain and range for the function. Let's study an easy example of a function f(x) = x + 1.
As soon as you possess the domain and the range for the function, you have to graph 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 leverage the horizontal line test. Immediately after you plot the graph of a function, draw horizontal lines over the graph. In the event that 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 does not intersect the graph at more than one place, we can also deduct all linear functions are one-to-one functions. Don’t forget 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 chart the values of x-coordinates and y-coordinates, you ought to consider 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 indicates 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 examine the graph 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 crosses multiple horizontal lines. Case in point, for either 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 opposite 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 is also a one-to-one function. The inverse of the function essentially reverses the function.
For Instance, 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 inverse of this function will deduct 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 qualities of an inverse one-to-one function are identical to any other one-to-one functions. This implies that the reverse of a one-to-one function will have one domain for each range and pass the horizontal line test.
How do you determine the inverse of a One-to-One Function?
Figuring out the inverse of a function is simple. 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.
Just like we discussed earlier, the inverse of a one-to-one function undoes the function. Considering the original output value required us to add 5 to each input value, the new output value will require us to deduct 5 from each input value.
One to One Function Practice Questions
Consider 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 whether or not the function is one-to-one.
2. Plot the function and its inverse.
3. Determine the inverse of the function numerically.
4. State the domain and range of both the function and its inverse.
5. Use the inverse to find the solution for x in each calculation.
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