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 corresponds to a single 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 the domain of the function, and the output value is noted as the range of the function.
Let's study the examples below:
For f(x), each value in the left circle correlates to a unique value in the right circle. Similarly, every value in the right circle corresponds to a unique value on the left. In mathematical words, this signifies every domain holds a unique range, and every range holds a unique domain. Thus, this is a representation of a one-to-one function.
Here are some more representations 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).
Notice 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 the same output, i.e., 4. In the same manner, the inputs -4 and 4 have identical output, i.e., 16. We can see that there are identical Y values for multiple X values. Thus, 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 the following qualities:
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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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They pass the horizontal line test.
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The graph of a function and its inverse are equivalent concerning the line y = x.
How to Graph a One to One Function
To graph a one-to-one function, you will need to figure out the domain and range for the function. Let's examine a straight-forward representation of a function f(x) = x + 1.
As soon as you possess the domain and the range for the function, you need to graph 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 test whether a function is one-to-one, we can use the horizontal line test. As soon as you chart the graph of a function, trace horizontal lines over the graph. In the event that a horizontal line intersects 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 deduct 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 study the graph for f(x) = x + 1. As soon as you graph the values for the x-coordinates and y-coordinates, you have to examine if 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 means that the function is a one-to-one function.
On the other hand, if the function is not a one-to-one function, it will intersect the same horizontal line more than once. Let's study the figure 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 meets various horizontal lines. For instance, for each domains -1 and 1, the range is 1. Additionally, for either -2 and 2, the range is 4. This means that f(x) = x^2 is not a one-to-one function.
What is the inverse of a One-to-One Function?
As a one-to-one function has a single 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 basically reverses the function.
For example, in the case of f(x) = x + 1, we add 1 to each value of x as a means of getting the output, or y. The inverse of this function will deduct 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 qualities of an inverse one-to-one function are the same as any other one-to-one functions. This implies that the inverse of a one-to-one function will have one domain for each range and pass the horizontal line test.
How do you find the inverse of a One-to-One Function?
Figuring out the inverse of a function is simple. You simply need 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.
As we reviewed previously, the inverse of a one-to-one function undoes the function. Because the original output value showed us we needed 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 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 each of these functions:
1. Identify whether or not the function is one-to-one.
2. Graph the function and its inverse.
3. Find the inverse of the function numerically.
4. Specify the domain and range of every function and its inverse.
5. Employ the inverse to solve for x in each equation.
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