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 whereby each input corresponds to just one output. In other words, for each x, there is a single 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 known as 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. In the same manner, any value in the right circle correlates to a unique value on the left. In mathematical jargon, this signifies every domain has a unique range, and every range owns 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 displays 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, in other words, 4. In conjunction, the inputs -4 and 4 have equal output, i.e., 16. We can comprehend that there are equivalent Y values for many 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 properties:
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The function holds 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 concerning the line y = x.
How to Graph a One to One Function
In order to graph a one-to-one function, you will have to find the domain and range for the function. Let's examine an easy representation of a function f(x) = x + 1.
Once you know 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 evaluate whether or not a Function is One to One?
To test whether a function is one-to-one, we can leverage the horizontal line test. Once you graph 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 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 place, 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 examine 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 point. 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 contrary, 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 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 intersects multiple horizontal lines. For example, for both domains -1 and 1, the range is 1. Additionally, for either -2 and 2, the range is 4. This implies 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 is also a one-to-one function. The opposite of the function basically reverses the function.
For Instance, in the example of f(x) = x + 1, we add 1 to each value of x for the purpose of getting the output, i.e., y. The opposite 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 qualities of an inverse one-to-one function are identical to every other one-to-one functions. This means 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 find the inverse of a One-to-One Function?
Determining the inverse of a function is not difficult. 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 discussed 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 subtract 5 from each input value.
One to One Function Practice Questions
Contemplate 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. Figure out if the function is one-to-one.
2. Plot the function and its inverse.
3. Find the inverse of the function numerically.
4. State the domain and range of both the function and its inverse.
5. Employ the inverse to find the solution for x in each formula.
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