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 correlates to just one output. In other words, for each x, there is just 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 noted as the domain of the function, and the output value is known as the range of the function.
Let's examine the images below:
For f(x), any value in the left circle corresponds to a unique value in the right circle. In conjunction, each value on the right correlates to a unique value on the left side. In mathematical terms, this signifies every domain holds a unique range, and every range owns a unique domain. Thus, 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 look at the second picture, which exhibits 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 example, the inputs -2 and 2 have equal output, that is, 4. In conjunction, the inputs -4 and 4 have equal output, i.e., 16. We can discern that there are matching Y values for numerous X values. Hence, 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 properties:
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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 equivalent regarding 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 determine the domain and range for the function. Let's examine an easy representation of a function f(x) = x + 1.
As soon as you know the domain and the range for the function, you have to graph the domain values on the X-axis and range values on the Y-axis.
How can you evaluate whether or not a Function is One to One?
To indicate whether or not a function is one-to-one, we can leverage 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 place, then the function is not one-to-one.
Because the graph of every linear function is a straight line, and a horizontal line does not intersect the graph at more than one spot, we can also deduct 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 examine the graph for f(x) = x + 1. As soon as you plot the values for the x-coordinates and y-coordinates, you ought to examine if a horizontal line intersects the graph at more than one place. 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 contrary, 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 instance, the graph intersects numerous horizontal lines. For example, for either domains -1 and 1, the range is 1. Additionally, for both -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 basically reverses 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 deduct 1 from each value of y.
The inverse of the function is known as f−1.
What are the qualities of the inverse of a One to One Function?
The qualities of an inverse one-to-one function are identical to every other one-to-one functions. This signifies that the inverse 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?
Determining the inverse of a function is simple. You simply have to change 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 reviewed earlier, 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 Examples
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 every function:
1. Identify if the function is one-to-one.
2. Chart the function and its inverse.
3. Figure out the inverse of the function numerically.
4. State the domain and range of each function and its inverse.
5. Use the inverse to find the solution for x in each calculation.
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