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 correlates to a single output. That is to say, for every x, there is only one 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 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 corresponds to a unique value in the right circle. In the same manner, every value in the right circle corresponds to a unique value in the left circle. In mathematical jargon, this means that every domain owns a unique range, and every range holds a unique domain. Therefore, this is a representation of a one-to-one function.
Here are some more representations 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 picture, which displays the values for g(x).
Be aware of the fact that the inputs in the left circle (domain) do not have unique outputs in the right circle (range). Case in point, the inputs -2 and 2 have equal output, i.e., 4. In conjunction, the inputs -4 and 4 have equal output, i.e., 16. We can comprehend that there are matching Y values for numerous X values. Therefore, 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 properties of One to One Functions?
One-to-one functions have these properties:
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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 the same concerning the line y = x.
How to Graph a One to One Function
When trying to graph a one-to-one function, you will need to figure out the domain and range for the function. Let's examine a simple representation 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 determine whether a Function is One to One?
To test if a function is one-to-one, we can use the horizontal line test. Immediately after you plot the graph of a function, draw horizontal lines over the graph. If a horizontal line passes through the graph of the function at more than one spot, then the function is not one-to-one.
Because the graph of every linear function is a straight line, and a horizontal line will not intersect the graph at more than one spot, we can also reason that all linear functions are one-to-one functions. Keep in mind that we do not leverage the vertical line test for one-to-one functions.
Let's look at the graph for f(x) = x + 1. Once you chart the values to x-coordinates and y-coordinates, you need to examine 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 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 examine 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 numerous horizontal lines. Case in point, for each domains -1 and 1, the range is 1. Similarly, for both -2 and 2, the range is 4. This signifies that f(x) = x^2 is not a one-to-one function.
What is the opposite of a One-to-One Function?
Considering the fact that 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 essentially undoes the function.
For Instance, in the case of f(x) = x + 1, we add 1 to each value of x for the purpose of getting the output, i.e., y. The inverse of this function will subtract 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 qualities of an inverse one-to-one function are the same as all other one-to-one functions. This implies that the reverse of a one-to-one function will hold one domain for each range and pass the horizontal line test.
How do you figure out the inverse of a One-to-One Function?
Finding the inverse of a function is simple. You just need to change the x and y values. Case in point, the inverse of the function f(x) = x + 5 is f-1(x) = x - 5.
Considering what we reviewed before, the inverse of a one-to-one function reverses the function. Since the original output value required us to add 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
Contemplate 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. Figure out whether or not the function is one-to-one.
2. Chart the function and its inverse.
3. Find the inverse of the function algebraically.
4. State the domain and range of each function and its inverse.
5. Use the inverse to solve for x in each equation.
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