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. In other words, for each 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 the domain of the function, and the output value is noted as the range of the function.
Let's study the pictures below:
For f(x), any value in the left circle corresponds to a unique value in the right circle. In the same manner, any value on the right corresponds to a unique value on the left. In mathematical words, this signifies every domain holds a unique range, and every range has a unique domain. Thus, this is an example 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 study the second image, which shows the values for g(x).
Be aware of the fact that the inputs in the left circle (domain) do not hold unique outputs in the right circle (range). For example, the inputs -2 and 2 have identical 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 numerous X values. Therefore, this is not a one-to-one function.
Here are different 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 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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It passes 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
To graph a one-to-one function, you will need to find the domain and range for the function. Let's look at a simple example of a function f(x) = x + 1.
Immediately after you possess the domain and the range for the function, you have to chart the domain values on the X-axis and range values on the Y-axis.
How can you determine if a Function is One to One?
To indicate whether or not a function is one-to-one, we can use the horizontal line test. As soon as you chart the graph of a function, trace horizontal lines over the graph. If a horizontal line moves through the graph of the function at more than one spot, 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 conclude all linear functions are one-to-one functions. Don’t forget that we do not use the vertical line test for one-to-one functions.
Let's study the graph for f(x) = x + 1. As soon as you chart the values of x-coordinates and y-coordinates, you ought to examine if 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 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 example, the graph intersects multiple horizontal lines. Case in point, for either domains -1 and 1, the range is 1. Similarly, 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?
As 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 basically reverses the function.
For Instance, in the case of f(x) = x + 1, we add 1 to each value of x as a means of getting the output, in other words, y. The opposite of this function will deduct 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 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 hold one domain for each range and pass the horizontal line test.
How do you determine the inverse of a One-to-One Function?
Finding the inverse of a function is simple. You just need to switch the x and y values. For example, the inverse of the function f(x) = x + 5 is f-1(x) = x - 5.
As we learned before, the inverse of a one-to-one function undoes the function. Considering the original output value required adding 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
Consider 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 any of these functions:
1. Identify whether the function is one-to-one.
2. Chart the function and its inverse.
3. Figure out the inverse of the function algebraically.
4. Specify the domain and range of every function and its inverse.
5. Employ the inverse to solve for x in each formula.
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