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 only one output. That is to say, for each 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 study the images below:
For f(x), any value in the left circle correlates to a unique value in the right circle. Similarly, each value on the right side corresponds to a unique value in the left circle. In mathematical terms, this means that every domain holds a unique range, and every range owns a unique domain. Therefore, this is a representation 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 displays the values for g(x).
Pay attention to the fact 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 equal output, i.e., 4. In conjunction, the inputs -4 and 4 have the same output, i.e., 16. We can discern that there are equivalent 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 properties of One to One Functions?
One-to-one functions have these 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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It passes the horizontal line test.
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The graph of a function and its inverse are the same with respect to the line y = x.
How to Graph a One to One Function
To graph a one-to-one function, you will have to figure out the domain and range for the function. Let's look at a straight-forward representation of a function f(x) = x + 1.
Immediately after you know the domain and the range for the function, you need 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 test if a function is one-to-one, we can leverage the horizontal line test. As soon as 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.
Since the graph of every linear function is a straight line, and a horizontal line doesn’t intersect the graph at more than one point, 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 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 consider whether 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 other hand, if the function is not a one-to-one function, it will intersect the same horizontal line more than once. Let's look at 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. Case in point, for both domains -1 and 1, the range is 1. In the same manner, for each -2 and 2, the range is 4. This means 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 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 inverse of the function essentially reverses the function.
For Instance, in the case of f(x) = x + 1, we add 1 to each value of x as a means of getting the output, i.e., y. The opposite 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 characteristics of an inverse one-to-one function are identical to any other one-to-one functions. This means that the inverse of a one-to-one function will hold one domain for each range and pass the horizontal line test.
How do you find the inverse of a One-to-One Function?
Finding the inverse of a function is simple. You simply have to swap 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 before, the inverse of a one-to-one function undoes the function. Since 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 Questions
Consider 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 any of these functions:
1. Figure out whether or not the function is one-to-one.
2. Draw the function and its inverse.
3. Determine the inverse of the function numerically.
4. State the domain and range of both the function and its inverse.
5. Apply the inverse to find the solution for x in each equation.
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