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 a single output. That is to say, for every 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 known as the domain of the function, and the output value is the range of the function.
Let's look at the examples below:
For f(x), every value in the left circle corresponds to a unique value in the right circle. Similarly, each value on the right correlates to a unique value on the left side. In mathematical terms, this means that every domain holds a unique range, and every range holds a unique domain. Therefore, this is an example of a one-to-one function.
Here are some additional examples of one-to-one functions:
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f(x) = x + 1
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f(x) = 2x
Now let's look at the second picture, which displays the values for g(x).
Be aware of the fact that the inputs in the left circle (domain) do not own unique outputs in the right circle (range). For example, the inputs -2 and 2 have the same output, in other words, 4. In conjunction, the inputs -4 and 4 have identical output, i.e., 16. We can comprehend that there are matching Y values for multiple X values. Hence, this is not a one-to-one function.
Here are some other 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 these properties:
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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 the same regarding the line y = x.
How to Graph a One to One Function
To graph a one-to-one function, you are required to figure out the domain and range for the function. Let's look at a straight-forward representation of a function f(x) = x + 1.
Immediately after you know 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 tell whether a Function is One to One?
To indicate whether or not a function is one-to-one, we can use the horizontal line test. Once 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 point, 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 doesn’t 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. Once you plot the values to x-coordinates and y-coordinates, you ought to consider if a horizontal line intersects the graph at more than one spot. 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 multiple times. 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 case, the graph crosses various horizontal lines. For instance, for both domains -1 and 1, the range is 1. Similarly, 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 opposite of a One-to-One Function?
Since 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 inverse of the function essentially reverses the function.
Case in point, in the case of f(x) = x + 1, we add 1 to each value of x as a means of getting the output, i.e., 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 characteristics of the inverse of a One to One Function?
The characteristics of an inverse one-to-one function are the same as all other one-to-one functions. This implies 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 determine the inverse of a One-to-One Function?
Figuring out the inverse of a function is not difficult. You just need to switch the x and y values. For example, the inverse of the function f(x) = x + 5 is f-1(x) = x - 5.
Considering what we discussed before, the inverse of a one-to-one function undoes the function. Because 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
Examine 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 any of these functions:
1. Figure out whether the function is one-to-one.
2. Graph the function and its inverse.
3. Determine the inverse of the function algebraically.
4. State the domain and range of both the function and its inverse.
5. Employ the inverse to solve for x in each formula.
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