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. So, for each x, there is a single 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 known as the range of the function.
Let's examine the pictures 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 words, this means that every domain holds a unique range, and every range has a unique domain. Therefore, this is a representation of a one-to-one function.
Here are some other examples of one-to-one functions:
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f(x) = x + 1
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f(x) = 2x
Now let's examine the second picture, which shows the values for g(x).
Be aware of the fact that the inputs in the left circle (domain) do not have unique outputs in the right circle (range). Case in point, the inputs -2 and 2 have equal output, that is, 4. Similarly, the inputs -4 and 4 have the same output, i.e., 16. We can discern that there are identical Y values for multiple 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 qualities 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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They pass 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
When trying to graph a one-to-one function, you will have to figure out the domain and range for the function. Let's look at a straight-forward example of a function f(x) = x + 1.
Immediately after you have the domain and the range for the function, you ought to graph the domain values on the X-axis and range values on the Y-axis.
How can you tell whether or not a Function is One to One?
To indicate whether a function is one-to-one, we can leverage the horizontal line test. Immediately after you graph the graph of a function, trace horizontal lines over the graph. If a horizontal line passes 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 deduct all linear functions are one-to-one functions. Remember 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 for the x-coordinates and y-coordinates, you have to consider 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 indicates 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 one time. 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 intersects multiple horizontal lines. For example, for either domains -1 and 1, the range is 1. Additionally, 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 opposite of a One-to-One Function?
Considering the fact that a one-to-one function has a single 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 undoes the function.
Case in point, in the case of f(x) = x + 1, we add 1 to each value of x in order to get the output, i.e., y. The opposite of this function will subtract 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 properties of an inverse one-to-one function are the same as every other one-to-one functions. This implies that the reverse of a one-to-one function will hold one domain for every 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 have to swap the x and y values. For example, the inverse of the function f(x) = x + 5 is f-1(x) = x - 5.
As we reviewed previously, the inverse of a one-to-one function reverses the function. Considering the original output value required us to add 5 to each input value, the new output value will require us to subtract 5 from each input value.
One to One Function Practice Questions
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 every function:
1. Identify whether or not the function is one-to-one.
2. Plot the function and its inverse.
3. Find the inverse of the function mathematically.
4. Specify the domain and range of each function and its inverse.
5. Use the inverse to determine the value for x in each equation.
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