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. So, for each x, there is just one 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 noted as the domain of the function, and the output value is known as the range of the function.
Let's examine the images below:
For f(x), each value in the left circle correlates to a unique value in the right circle. In conjunction, every value on the right side corresponds to a unique value on the left. In mathematical words, this implies every domain has a unique range, and every range owns a unique domain. Hence, this is an example of a one-to-one function.
Here are some different 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 picture, 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 example, the inputs -2 and 2 have equal output, i.e., 4. Similarly, the inputs -4 and 4 have identical output, i.e., 16. We can comprehend that there are equivalent Y values for numerous X values. Therefore, 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 properties of One to One Functions?
One-to-one functions have the following characteristics:
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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 with respect to the line y = x.
How to Graph a One to One Function
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 simple example of a function f(x) = x + 1.
Once you possess the domain and the range for the function, you have to graph 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 a function is one-to-one, we can use the horizontal line test. Immediately after you graph the graph of a function, trace 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.
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 conclude all linear functions are one-to-one functions. Keep in mind that we do not leverage the vertical line test for one-to-one functions.
Let's look at the graph for f(x) = x + 1. As soon as you plot the values of x-coordinates and y-coordinates, you ought to examine whether a horizontal line intersects the graph at more than one place. In this case, 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 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 case, the graph crosses multiple horizontal lines. For example, for each 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?
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 is also a one-to-one function. The opposite of the function essentially undoes the function.
For example, 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 subtract 1 from each value of y.
The inverse of the function is denoted 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 means that the reverse of a one-to-one function will hold one domain for each range and pass the horizontal line test.
How do you find the inverse of a One-to-One Function?
Finding the inverse of a function is simple. You just have to change the x and y values. For instance, the inverse of the function f(x) = x + 5 is f-1(x) = x - 5.
Just like we learned previously, the inverse of a one-to-one function undoes the function. Since the original output value showed us we needed to add 5 to each input value, the new output value will require us to deduct 5 from each input value.
One to One Function Practice Questions
Contemplate these 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. Identify if the function is one-to-one.
2. Chart the function and its inverse.
3. Find the inverse of the function mathematically.
4. State the domain and range of each function and its inverse.
5. Apply the inverse to find the solution for x in each calculation.
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