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 only one output. In other words, 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 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), each value in the left circle correlates to a unique value in the right circle. In conjunction, each value in the right circle 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 different examples 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 shows 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 instance, the inputs -2 and 2 have the same output, i.e., 4. In the same manner, the inputs -4 and 4 have equal output, i.e., 16. We can discern that there are equivalent Y values for many X values. Therefore, this is not a one-to-one function.
Here are additional 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 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 identical 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 examine a straight-forward example of a function f(x) = x + 1.
Immediately after you know the domain and the range for the function, you ought to chart the domain values on the X-axis and range values on the Y-axis.
How can you tell if a Function is One to One?
To test whether a function is one-to-one, we can apply the horizontal line test. As soon as you plot the graph of a function, trace horizontal lines over the graph. In the event that 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 place, we can also conclude all linear functions are one-to-one functions. Keep in mind 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 chart the values of x-coordinates and y-coordinates, you have to review if 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 one time. 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 instance, the graph meets numerous horizontal lines. Case in point, for each domains -1 and 1, the range is 1. In the same manner, for both -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 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 basically 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, in other words, y. The inverse of this function will deduct 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 properties of an inverse one-to-one function are the same as every other one-to-one functions. This signifies that the opposite of a one-to-one function will hold one domain for every range and pass the horizontal line test.
How do you determine the inverse of a One-to-One Function?
Finding the inverse of a function is not difficult. You simply have to swap 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 learned 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 Questions
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 any 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 algebraically.
4. State the domain and range of every function and its inverse.
5. Use the inverse to find the solution for x in each equation.
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