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 whereby each input corresponds to a single output. In other words, for each x, there is a single 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 noted as the domain of the function, and the output value is 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. In the same manner, any value in the right circle corresponds to a unique value in the left circle. In mathematical jargon, this signifies 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 other 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 picture, which shows the values for g(x).
Notice 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, that is, 4. Similarly, the inputs -4 and 4 have the same output, i.e., 16. We can see that there are matching Y values for many X values. Thus, this is not a one-to-one function.
Here are different 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 qualities of One to One Functions?
One-to-one functions have these qualities:
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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 the same regarding the line y = x.
How to Graph a One to One Function
In order to graph a one-to-one function, you are required to find the domain and range for the function. Let's study an easy representation of a function f(x) = x + 1.
Immediately after you have 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 a Function is One to One?
To test whether 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 point, then the function is not one-to-one.
Because the graph of every linear function is a straight line, and a horizontal line will not intersect the graph at more than one place, we can also conclude all linear functions are one-to-one functions. Keep in mind that we do not apply the vertical line test for one-to-one functions.
Let's look at the graph for f(x) = x + 1. Immediately after you plot the values to x-coordinates and y-coordinates, you ought to review whether or not 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 means 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 multiple times. Let's look at 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 case, the graph intersects various horizontal lines. For instance, for either domains -1 and 1, the range is 1. Additionally, for either -2 and 2, the range is 4. This implies 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 inverse of the function essentially reverses the function.
For example, in the event of f(x) = x + 1, we add 1 to each value of x as a means of getting the output, i.e., y. The opposite of this function will remove 1 from each value of y.
The inverse of the function is denoted as f−1.
What are the properties of the inverse of a One to One Function?
The qualities of an inverse one-to-one function are no different than every other one-to-one functions. This implies that the reverse of a one-to-one function will have one domain for every 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 very easy. You simply need to change 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 previously, the inverse of a one-to-one function reverses the function. Considering the original output value showed us we needed to add 5 to each input value, the new output value will require us to subtract 5 from each input value.
One to One Function Practice Questions
Contemplate these 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. Figure out if the function is one-to-one.
2. Plot the function and its inverse.
3. Determine the inverse of the function algebraically.
4. Specify the domain and range of every function and its inverse.
5. Employ the inverse to determine the value for x in each calculation.
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