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 only one output. That is to say, 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 known as the domain of the function, and the output value is known as the range of the function.
Let's examine the examples below:
For f(x), any value in the left circle corresponds to a unique value in the right circle. In conjunction, each value on the right side corresponds to a unique value on the left. In mathematical terms, this implies every domain owns a unique range, and every range has a unique domain. Therefore, this is an example of a one-to-one function.
Here are some additional 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 exhibits 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 instance, the inputs -2 and 2 have the same output, i.e., 4. In the same manner, the inputs -4 and 4 have equal output, i.e., 16. We can discern that there are identical Y values for numerous X values. Hence, this is not a one-to-one function.
Here are additional 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 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 identical regarding the line y = x.
How to Graph a One to One Function
In order to graph a one-to-one function, you will need to find the domain and range for the function. Let's study a simple example of a function f(x) = x + 1.
As soon as you know the domain and the range for the function, you need to plot the domain values on the X-axis and range values on the Y-axis.
How can you evaluate whether a Function is One to One?
To test 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 moves through the graph of the function at more than one place, 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 does not intersect the graph at more than one spot, we can also conclude all linear functions are one-to-one functions. Don’t forget 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. As soon as you chart the values of x-coordinates and y-coordinates, you have to review if a horizontal line intersects the graph at more than one spot. 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 study 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 instance, the graph crosses numerous 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?
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 inverse of the function essentially reverses the function.
For Instance, 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 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 properties of the inverse of a One to One Function?
The characteristics 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 possess one domain for every 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 very easy. You simply have to swap 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 reverses the function. Since 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 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. Chart the function and its inverse.
3. Find the inverse of the function algebraically.
4. State the domain and range of every function and its inverse.
5. Use the inverse to solve for x in each equation.
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