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. So, for each x, there is only one 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 noted as the domain of the function, and the output value is the range of the function.
Let's study the pictures below:
For f(x), every value in the left circle corresponds to a unique value in the right circle. In the same manner, each value on the right side corresponds to a unique value on the left. In mathematical words, this means that every domain owns a unique range, and every range owns a unique domain. Therefore, this is a representation 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 look at the second example, which displays 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 instance, the inputs -2 and 2 have the same output, i.e., 4. In the same manner, the inputs -4 and 4 have the same output, i.e., 16. We can discern that there are equivalent Y values for many X values. Thus, 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 characteristics of One to One Functions?
One-to-one functions have these characteristics:
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The function holds 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 equivalent 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 determine the domain and range for the function. Let's study an easy representation of a function f(x) = x + 1.
As soon as you possess the domain and the range for the function, you ought to plot 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 prove whether or not a function is one-to-one, we can use the horizontal line test. As soon as you plot the graph of a function, trace horizontal lines over the graph. If a horizontal line intersects 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 will not intersect the graph at more than one point, we can also conclude all linear functions are one-to-one functions. Remember that we do not apply the vertical line test for one-to-one functions.
Let's examine the graph for f(x) = x + 1. Immediately after you graph the values for the x-coordinates and y-coordinates, you need 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 signifies 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 more than once. Let's examine 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 various 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 implies that f(x) = x^2 is not a one-to-one function.
What is the inverse 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 also happens to be a one-to-one function. The opposite of the function essentially reverses the function.
For example, in the event of f(x) = x + 1, we add 1 to each value of x as a means 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 characteristics of the inverse of a One to One Function?
The qualities of an inverse one-to-one function are no different than any other one-to-one functions. This signifies that the inverse of a one-to-one function will possess one domain for each range and pass the horizontal line test.
How do you find the inverse of a One-to-One Function?
Determining the inverse of a function is simple. You just 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 learned 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 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 every function:
1. Determine whether or not the function is one-to-one.
2. Draw the function and its inverse.
3. Determine the inverse of the function algebraically.
4. Specify the domain and range of every function and its inverse.
5. Apply the inverse to find the solution for x in each formula.
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