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 correlates to just one output. That is to say, 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 noted as 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, each value on the right corresponds to a unique value on the left. In mathematical jargon, this signifies every domain has a unique range, and every range holds a unique domain. Hence, 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 picture, which shows the values for g(x).
Pay attention to the fact that the inputs in the left circle (domain) do not have unique outputs in the right circle (range). For instance, the inputs -2 and 2 have equal output, i.e., 4. Similarly, the inputs -4 and 4 have the same output, i.e., 16. We can discern that there are equivalent Y values for numerous X values. Thus, 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 qualities of One to One Functions?
One-to-one functions have these 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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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
When trying to graph a one-to-one function, you are required to determine the domain and range for the function. Let's examine a straight-forward example of a function f(x) = x + 1.
Immediately after you have the domain and the range for the function, you need to plot the domain values on the X-axis and range values on the Y-axis.
How can you determine if a Function is One to One?
To test whether or not a function is one-to-one, we can use the horizontal line test. As soon as you graph the graph of a function, draw horizontal lines over the graph. In the event that a horizontal line passes through the graph of the function at more than one place, then the function is not one-to-one.
Since the graph of every linear function is a straight line, and a horizontal line does not intersect the graph at more than one point, we can also deduct all linear functions are one-to-one functions. Keep in mind that we do not apply the vertical line test for one-to-one functions.
Let's examine the graph for f(x) = x + 1. Once you chart the values for the x-coordinates and y-coordinates, you ought to review whether or not a horizontal line intersects the graph at more than one spot. In this example, 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 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 various horizontal lines. Case in point, for either domains -1 and 1, the range is 1. Additionally, for either -2 and 2, the range is 4. This signifies that f(x) = x^2 is not a one-to-one function.
What is the inverse of a One-to-One Function?
Considering the fact that 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 opposite of the function essentially reverses the function.
For example, in the example of f(x) = x + 1, we add 1 to each value of x for the purpose of getting the output, or y. The inverse of this function will subtract 1 from each value of y.
The inverse of the function is f−1.
What are the qualities of the inverse of a One to One Function?
The qualities of an inverse one-to-one function are no different than all other one-to-one functions. This signifies that the reverse 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. For instance, the inverse of the function f(x) = x + 5 is f-1(x) = x - 5.
As we reviewed before, the inverse of a one-to-one function reverses the function. Because the original output value required adding 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
Examine 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 whether the function is one-to-one.
2. Graph the function and its inverse.
3. Figure out the inverse of the function numerically.
4. Specify the domain and range of each function and its inverse.
5. Use the inverse to solve for x in each formula.
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