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 just one output. In other words, for each 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 noted as the domain of the function, and the output value is known as the range of the function.
Let's examine the pictures 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. In mathematical words, this signifies every domain has a unique range, and every range holds a unique domain. Therefore, this is a representation of a one-to-one function.
Here are some different examples 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 displays 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 instance, the inputs -2 and 2 have equal output, in other words, 4. In conjunction, the inputs -4 and 4 have identical output, i.e., 16. We can discern that there are equivalent Y values for multiple X values. Therefore, this is not a one-to-one function.
Here are additional 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 qualities of One to One Functions?
One-to-one functions have these qualities:
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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 identical 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 are required to determine the domain and range for the function. Let's study an easy representation 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 indicate whether a function is one-to-one, we can apply the horizontal line test. Once you graph 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 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 place, we can also reason that 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 examine the graph for f(x) = x + 1. Immediately after you chart the values for the x-coordinates and y-coordinates, you ought to review whether or not a horizontal line intersects the graph at more than one place. In this instance, 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 one time. Let's study 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 each domains -1 and 1, the range is 1. Additionally, for each -2 and 2, the range is 4. This signifies 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 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 basically undoes the function.
For example, 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 deduct 1 from each value of y.
The inverse of the function is known 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 each range and pass the horizontal line test.
How do you find the inverse of a One-to-One Function?
Finding the inverse of a function is not difficult. You simply need to swap the x and y values. Case in point, the inverse of the function f(x) = x + 5 is f-1(x) = x - 5.
As we discussed earlier, the inverse of a one-to-one function undoes the function. Since the original output value required us 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 whether the function is one-to-one.
2. Plot the function and its inverse.
3. Determine the inverse of the function numerically.
4. Specify the domain and range of every function and its inverse.
5. Use the inverse to solve for x in each formula.
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