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 corresponds to a single output. So, for every 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 known as the range of the function.
Let's examine the examples below:
For f(x), any value in the left circle corresponds 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 signifies every domain holds a unique range, and every range holds a unique domain. Hence, this is an example of a one-to-one function.
Here are some other 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 image, 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). For instance, the inputs -2 and 2 have the same output, in other words, 4. In the same manner, the inputs -4 and 4 have the same output, i.e., 16. We can discern that there are identical Y values for numerous X values. Thus, 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 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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They pass the horizontal line test.
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The graph of a function and its inverse are equivalent concerning 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 figure out the domain and range for the function. Let's look at a straight-forward example of a function f(x) = x + 1.
As soon as you have the domain and the range for the function, you have 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 if a function is one-to-one, we can apply the horizontal line test. Immediately after you chart 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 spot, 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 place, we can also reason that 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 plot the values to x-coordinates and y-coordinates, you need to examine if a horizontal line intersects the graph at more than one point. In this case, the graph does not intersect any horizontal line more than once. This signifies 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 more than one time. 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 instance, the graph crosses multiple horizontal lines. For example, for both domains -1 and 1, the range is 1. In the same manner, 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 opposite of a One-to-One Function?
As 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 inverse of the function basically undoes the function.
For Instance, in the event 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 remove 1 from each value of y.
The inverse of the function is denoted as f−1.
What are the properties of the inverse of a One to One Function?
The properties of an inverse one-to-one function are the same as all other one-to-one functions. This implies that the opposite of a one-to-one function will possess one domain for every 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 simple. You just have 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 reviewed before, 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 subtract 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. Chart the function and its inverse.
3. Find the inverse of the function numerically.
4. Specify the domain and range of every function and its inverse.
5. Employ the inverse to find the solution for x in each equation.
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