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 corresponds to just one output. In other words, for every x, there is only one y and vice versa. This means 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 examine the pictures below:
For f(x), any value in the left circle correlates to a unique value in the right circle. In the same manner, each value in the right circle corresponds to a unique value in the left circle. In mathematical terms, this signifies every domain has a unique range, and every range owns a unique domain. Therefore, this is an example of a one-to-one function.
Here are some additional representations 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 image, which exhibits the values for g(x).
Notice that the inputs in the left circle (domain) do not hold unique outputs in the right circle (range). For example, the inputs -2 and 2 have identical output, in other words, 4. In conjunction, the inputs -4 and 4 have identical output, i.e., 16. We can comprehend that there are matching Y values for many X values. Hence, this is not a one-to-one function.
Here are additional 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 properties 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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The function 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
To graph a one-to-one function, you will need to determine the domain and range for the function. Let's look at a straight-forward example of a function f(x) = x + 1.
Once you know the domain and the range for the function, you ought to plot the domain values on the X-axis and range values on the Y-axis.
How can you tell whether or not a Function is One to One?
To indicate if a function is one-to-one, we can use the horizontal line test. Immediately after you chart the graph of a function, trace horizontal lines over the graph. In the event that a horizontal line passes 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 deduct all linear functions are one-to-one functions. Don’t forget 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 to x-coordinates and y-coordinates, you have 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 means 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 look at the diagram 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 various horizontal lines. For example, for either domains -1 and 1, the range is 1. Similarly, 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 opposite 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 reverses 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, i.e., 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 characteristics of the inverse of a One to One Function?
The characteristics of an inverse one-to-one function are identical to any other one-to-one functions. This signifies that the reverse of a one-to-one function will possess 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 not difficult. You simply have to switch the x and y values. Case in point, the inverse of the function f(x) = x + 5 is f-1(x) = x - 5.
As we discussed earlier, 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 deduct 5 from each input value.
One to One Function Practice Examples
Examine 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 every function:
1. Figure out whether the function is one-to-one.
2. Graph the function and its inverse.
3. Figure out the inverse of the function numerically.
4. Indicate the domain and range of each function and its inverse.
5. Use the inverse to determine the value for x in each equation.
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