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 where each input corresponds to just one output. That is to say, for every 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 pictures below:
For f(x), each value in the left circle corresponds to a unique value in the right circle. Similarly, every value in the right circle corresponds to a unique value in the left circle. In mathematical jargon, this implies every domain holds a unique range, and every range holds a unique domain. Thus, 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 shows the values for g(x).
Notice that the inputs in the left circle (domain) do not own unique outputs in the right circle (range). Case in point, the inputs -2 and 2 have identical output, i.e., 4. Similarly, the inputs -4 and 4 have the same output, i.e., 16. We can discern that there are matching Y values for multiple X values. Therefore, this is not a one-to-one function.
Here are different 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 characteristics of One to One Functions?
One-to-one functions have these characteristics:
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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
In order to graph a one-to-one function, you will need to figure out the domain and range for the function. Let's look at an easy representation of a function f(x) = x + 1.
As soon as you know 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 evaluate if a Function is One to One?
To test whether or not a function is one-to-one, we can apply the horizontal line test. As soon as you plot 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 point, 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 place, we can also reason that 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 examine the graph for f(x) = x + 1. Once you graph the values of x-coordinates and y-coordinates, you ought to review if a horizontal line intersects the graph at more than one spot. In this case, the graph does not intersect any horizontal line more than once. This indicates that the function is a one-to-one function.
On the other hand, if the function is not a one-to-one function, it will intersect the same horizontal line multiple times. Let's study 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 example, the graph meets numerous horizontal lines. For example, for both domains -1 and 1, the range is 1. In the same manner, for either -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?
Considering the fact that 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.
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 inverse of this function will remove 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 characteristics 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 determine the inverse of a One-to-One Function?
Finding the inverse of a function is simple. You just 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.
As we discussed before, the inverse of a one-to-one function reverses the function. Considering the original output value required us to add 5 to each input value, the new output value will require us to delete 5 from each input value.
One to One Function Practice Questions
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 each of these functions:
1. Determine whether or not the function is one-to-one.
2. Plot the function and its inverse.
3. Figure out the inverse of the function numerically.
4. State the domain and range of both the function and its inverse.
5. Apply the inverse to solve for x in each equation.
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