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 in which each input correlates to a single output. So, for every x, there is just 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 examples below:
For f(x), any value in the left circle correlates to a unique value in the right circle. In conjunction, every value on the right correlates to a unique value in the left circle. In mathematical terms, this signifies every domain holds a unique range, and every range has a unique domain. Hence, this is an example of a one-to-one function.
Here are some additional 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 exhibits the values for g(x).
Be aware of the fact that the inputs in the left circle (domain) do not own unique outputs in the right circle (range). For instance, the inputs -2 and 2 have equal output, that is, 4. Similarly, the inputs -4 and 4 have identical output, i.e., 16. We can see that there are identical Y values for many X values. Thus, 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 characteristics of One to One Functions?
One-to-one functions have these qualities:
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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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The function passes the horizontal line test.
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The graph of a function and its inverse are the same regarding the line y = x.
How to Graph a One to One Function
To graph a one-to-one function, you will need to determine the domain and range for the function. Let's study a simple representation of a function f(x) = x + 1.
Once you possess the domain and the range for the function, you have to plot 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 prove whether or not a function is one-to-one, we can use the horizontal line test. As soon as you graph the graph of a function, draw horizontal lines over the graph. If a horizontal line intersects the graph of the function at more than one spot, then the function is not one-to-one.
Due to the fact that the graph of every linear function is a straight line, and a horizontal line will not intersect the graph at more than one spot, we can also reason that 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 look at the graph for f(x) = x + 1. As soon as you chart the values for the x-coordinates and y-coordinates, you have to consider whether a horizontal line intersects the graph at more than one place. In this example, 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 more than once. 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 numerous horizontal lines. Case in point, for either domains -1 and 1, the range is 1. Additionally, 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 inverse of a One-to-One Function?
Since a one-to-one function has only one input value for each output value, the inverse of a one-to-one function is also a one-to-one function. The opposite of the function basically reverses the function.
Case in point, in the case of f(x) = x + 1, we add 1 to each value of x for the purpose of getting the output, or y. The opposite of this function will remove 1 from each value of y.
The inverse of the function is f−1.
What are the qualities of the inverse of a One to One Function?
The characteristics 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 possess one domain for each range and pass the horizontal line test.
How do you figure out the inverse of a One-to-One Function?
Determining the inverse of a function is very easy. 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.
Just like we discussed 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 subtract 5 from each input value.
One to One Function Practice Examples
Contemplate 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. Determine whether the function is one-to-one.
2. Chart the function and its inverse.
3. Figure out the inverse of the function mathematically.
4. Indicate 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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