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 correlates to only one output. In other words, for each x, there is only one 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 the domain of the function, and the output value is the range of the function.
Let's look at the pictures below:
For f(x), every value in the left circle correlates to a unique value in the right circle. Similarly, any value on the right side corresponds to a unique value on the left. In mathematical jargon, this signifies 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 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 look at 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). For example, the inputs -2 and 2 have equal output, that is, 4. Similarly, the inputs -4 and 4 have equal output, i.e., 16. We can comprehend that there are matching Y values for multiple X values. Hence, 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 characteristics of One to One Functions?
One-to-one functions have these properties:
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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 identical concerning 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 straight-forward representation of a function f(x) = x + 1.
Once 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 whether or not a Function is One to One?
To test if a function is one-to-one, we can leverage the horizontal line test. Immediately after you plot the graph of a function, trace horizontal lines over the graph. If a horizontal line intersects the graph of the function at more than one place, then the function is not one-to-one.
Because the graph of every linear function is a straight line, and a horizontal line doesn’t 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 examine the graph for f(x) = x + 1. Once you graph the values of x-coordinates and y-coordinates, you ought to review whether 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 indicates 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 diagram 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 meets multiple horizontal lines. For example, for both 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 opposite 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 inverse of the function basically undoes the function.
Case in point, in the example of f(x) = x + 1, we add 1 to each value of x in order to get the output, in other words, y. The inverse 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 qualities of an inverse one-to-one function are no different than every other one-to-one functions. This implies that the inverse of a one-to-one function will hold 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 very easy. You simply have to change 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 learned earlier, 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 subtract 5 from each input value.
One to One Function Practice Examples
Examine 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 any of these functions:
1. Figure out whether the function is one-to-one.
2. Graph the function and its inverse.
3. Find the inverse of the function algebraically.
4. Specify the domain and range of both the function and its inverse.
5. Use the inverse to determine the value for x in each equation.
Grade Potential Can Help You Master You Functions
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