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 correlates to just one output. That is to say, for each x, there is a single y and vice versa. This implies 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 look at the images below:
For f(x), each value in the left circle corresponds to a unique value in the right circle. In the same manner, any value on the right corresponds to a unique value in the left circle. In mathematical words, this signifies every domain owns a unique range, and every range has a unique domain. Therefore, this is an example 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 image, which displays 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 example, the inputs -2 and 2 have equal output, in other words, 4. In the same manner, the inputs -4 and 4 have the same output, i.e., 16. We can see that there are matching Y values for multiple X values. Hence, this is not a one-to-one function.
Here are some other 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 properties of One to One Functions?
One-to-one functions have the following characteristics:
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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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The function passes the horizontal line test.
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The graph of a function and its inverse are equivalent regarding the line y = x.
How to Graph a One to One Function
To graph a one-to-one function, you will need to determine the domain and range for the function. Let's study a simple representation of a function f(x) = x + 1.
As soon as you know 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 evaluate whether a Function is One to One?
To test whether a function is one-to-one, we can use the horizontal line test. Once you graph the graph of a function, draw horizontal lines over the graph. In the event that a horizontal line moves through 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 doesn’t intersect the graph at more than one spot, 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 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 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 means that the function is a one-to-one function.
Subsequently, if the function is not a one-to-one function, it will intersect the same horizontal line multiple times. Let's look at 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 intersects numerous horizontal lines. For example, for both domains -1 and 1, the range is 1. Similarly, for each -2 and 2, the range is 4. This signifies 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 only one input value for each output value, the inverse of a one-to-one function also happens to be a one-to-one function. The opposite of the function essentially undoes the function.
Case in point, in the case 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 deduct 1 from each value of y.
The inverse of the function is f−1.
What are the characteristics of the inverse of a One to One Function?
The qualities of an inverse one-to-one function are identical to all other one-to-one functions. This signifies that the opposite of a one-to-one function will possess 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 very easy. You simply need to change the x and y values. For instance, 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. Because 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 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. Determine whether the function is one-to-one.
2. Plot the function and its inverse.
3. Determine the inverse of the function numerically.
4. State the domain and range of each function and its inverse.
5. Employ the inverse to find the solution for x in each calculation.
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