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 correlates to just one output. So, for each 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 noted as the range of the function.
Let's examine the pictures below:
For f(x), each value in the left circle correlates to a unique value in the right circle. In the same manner, every value on the right correlates to a unique value in the left circle. In mathematical jargon, this implies 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 more 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 example, 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). For instance, the inputs -2 and 2 have identical output, that is, 4. Similarly, the inputs -4 and 4 have equal 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 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 qualities of One to One Functions?
One-to-one functions have the following 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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The function passes 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
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 example of a function f(x) = x + 1.
Once you possess 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 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 spot, 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 spot, we can also deduct 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. Immediately after you plot the values for the 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 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 figure 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 intersects multiple horizontal lines. For instance, for either domains -1 and 1, the range is 1. Similarly, for either -2 and 2, the range is 4. This signifies 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 just one input value for each output value, the inverse of a one-to-one function is also a one-to-one function. The opposite of the function basically reverses the function.
Case in point, in the event of f(x) = x + 1, we add 1 to each value of x in order to get the output, i.e., y. The opposite of this function will subtract 1 from each value of y.
The inverse of the function is denoted as 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 implies that the opposite of a one-to-one function will hold one domain for every 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 very easy. You simply need 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.
Just like we learned earlier, 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 subtract 5 from each input value.
One to One Function Practice Examples
Contemplate the following 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. Identify whether the function is one-to-one.
2. Draw the function and its inverse.
3. Find the inverse of the function numerically.
4. Indicate the domain and range of both the function and its inverse.
5. Apply the inverse to solve for x in each calculation.
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