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 a single output. In other words, for every 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 look at the pictures below:
For f(x), each value in the left circle corresponds to a unique value in the right circle. In conjunction, every value in the right circle correlates to a unique value on the left side. In mathematical terms, this means that every domain holds a unique range, and every range holds a unique domain. Hence, this is a representation of a one-to-one function.
Here are some more examples of one-to-one functions:
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f(x) = x + 1
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f(x) = 2x
Now let's look at the second picture, which displays the values for g(x).
Notice that the inputs in the left circle (domain) do not have unique outputs in the right circle (range). For example, the inputs -2 and 2 have the same output, in other words, 4. In conjunction, the inputs -4 and 4 have identical output, i.e., 16. We can see that there are matching Y values for many X values. Therefore, this is not a one-to-one function.
Here are some other 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 the following properties:
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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 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 figure out the domain and range for the function. Let's look at a simple example of a function f(x) = x + 1.
Immediately after 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 determine whether a Function is One to One?
To test whether a function is one-to-one, we can use the horizontal line test. Immediately after you chart 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 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 point, we can also reason that 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 study the graph for f(x) = x + 1. As soon as you graph the values of 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 means that the function is a one-to-one function.
On the contrary, if the function is not a one-to-one function, it will intersect the same horizontal line multiple times. 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. For instance, for each domains -1 and 1, the range is 1. In the same manner, 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 inverse 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 opposite of the function essentially reverses the function.
For example, in the case of f(x) = x + 1, we add 1 to each value of x for the purpose of getting the output, i.e., 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 inverse 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?
Determining the inverse of a function is not difficult. You simply need to switch the x and y values. Case in point, the inverse of the function f(x) = x + 5 is f-1(x) = x - 5.
Considering what we learned previously, the inverse of a one-to-one function undoes the function. Because the original output value required adding 5 to each input value, the new output value will require us to delete 5 from each input value.
One to One Function Practice Questions
Consider 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 each of these functions:
1. Figure out if the function is one-to-one.
2. Chart the function and its inverse.
3. Find 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 formula.
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