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 each x, there is only one 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 noted as the range of the function.
Let's look at the pictures below:
For f(x), any value in the left circle corresponds to a unique value in the right circle. In conjunction, any value on the right side corresponds to a unique value on the left. In mathematical jargon, this implies every domain has a unique range, and every range holds a unique domain. Thus, this is an example 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 study the second image, which displays the values for g(x).
Notice that the inputs in the left circle (domain) do not own unique outputs in the right circle (range). For example, 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 see that there are identical Y values for numerous X values. Therefore, 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 characteristics:
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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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It passes the horizontal line test.
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The graph of a function and its inverse are equivalent with respect to 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 examine a straight-forward example of a function f(x) = x + 1.
Immediately after you possess the domain and the range for the function, you need 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 indicate whether or not a function is one-to-one, we can leverage the horizontal line test. Once you plot the graph of a function, trace horizontal lines over the graph. In the event that a horizontal line passes 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 conclude all linear functions are one-to-one functions. Remember 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. Once you graph the values for the x-coordinates and y-coordinates, you need to consider whether or not 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 signifies 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 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 crosses numerous horizontal lines. Case in point, for either 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 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 opposite of the function essentially undoes the function.
For example, in the case 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 properties of the inverse of a One to One Function?
The properties of an inverse one-to-one function are the same as any other one-to-one functions. This signifies that the opposite of a one-to-one function will possess one domain for each range and pass the horizontal line test.
How do you find the inverse of a One-to-One Function?
Determining the inverse of a function is not difficult. You just have to switch the x and y values. For instance, the inverse of the function f(x) = x + 5 is f-1(x) = x - 5.
Considering what we learned earlier, the inverse of a one-to-one function undoes the function. Since the original output value required adding 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 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 the function is one-to-one.
2. Plot the function and its inverse.
3. Find the inverse of the function numerically.
4. Specify the domain and range of both the function and its inverse.
5. Employ the inverse to determine the value for x in each equation.
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