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 correlates to a single output. That is to say, for every 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 noted as the domain of the function, and the output value is noted as the range of the function.
Let's look at 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, any value on the right side corresponds to a unique value on the left. In mathematical words, this means that every domain has a unique range, and every range holds a unique domain. Thus, this is a representation of a one-to-one function.
Here are some more 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 exhibits 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, in other words, 4. In conjunction, the inputs -4 and 4 have the same output, i.e., 16. We can comprehend that there are matching Y values for many X values. Hence, this is not a one-to-one function.
Here are additional 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 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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The function passes the horizontal line test.
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The graph of a function and its inverse are identical regarding the line y = x.
How to Graph a One to One Function
To graph a one-to-one function, you are required to find the domain and range for the function. Let's look at a simple representation of a function f(x) = x + 1.
Once 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 tell whether a Function is One to One?
To prove if a function is one-to-one, we can leverage the horizontal line test. As soon as you graph 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 will not intersect the graph at more than one point, we can also deduct all linear functions are one-to-one functions. Remember that we do not use the vertical line test for one-to-one functions.
Let's study the graph for f(x) = x + 1. Once you plot the values to x-coordinates and y-coordinates, you ought to examine whether a horizontal line intersects the graph at more than one point. In this example, the graph does not intersect any horizontal line more than once. This signifies 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 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 instance, the graph intersects various horizontal lines. Case in point, for both domains -1 and 1, the range is 1. Similarly, for both -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 just 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 inverse of the function basically reverses the function.
For Instance, in the event of f(x) = x + 1, we add 1 to each value of x for the purpose of getting the output, in other words, 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 characteristics of the inverse of a One to One Function?
The properties of an inverse one-to-one function are no different than any other one-to-one functions. This means that the opposite 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 not difficult. You just 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.
Just like we discussed previously, the inverse of a one-to-one function undoes the function. Since the original output value showed us we needed 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 Examples
Examine 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 each of these functions:
1. Figure out if the function is one-to-one.
2. Graph the function and its inverse.
3. Determine the inverse of the function mathematically.
4. Specify the domain and range of both the function and its inverse.
5. Employ the inverse to solve for x in each formula.
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