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 a single output. So, for every 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 noted as the range of the function.
Let's study the images below:
For f(x), every value in the left circle corresponds to a unique value in the right circle. Similarly, every value on the right correlates to a unique value on the left side. In mathematical words, this means that 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 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 examine the second image, which displays the values for g(x).
Notice that the inputs in the left circle (domain) do not have unique outputs in the right circle (range). For example, the inputs -2 and 2 have the same output, i.e., 4. In conjunction, the inputs -4 and 4 have equal output, i.e., 16. We can discern that there are equivalent Y values for multiple 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 characteristics of One to One Functions?
One-to-one functions have these qualities:
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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 identical concerning 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 find the domain and range for the function. Let's study an easy representation of a function f(x) = x + 1.
Immediately after you know the domain and the range for the function, you need to chart 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 prove if a function is one-to-one, we can apply the horizontal line test. As soon as you chart the graph of a function, trace horizontal lines over the graph. In the event that a horizontal line moves 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 doesn’t intersect the graph at more than one spot, we can also deduct all linear functions are one-to-one functions. Don’t forget that we do not leverage the vertical line test for one-to-one functions.
Let's study the graph for f(x) = x + 1. As soon as you plot the values for the x-coordinates and y-coordinates, you have to examine if a horizontal line intersects the graph at more than one spot. In this instance, 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 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 example, the graph meets 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 means 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 is also a one-to-one function. The inverse 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 as a means of getting the output, or y. The opposite of this function will remove 1 from each value of y.
The inverse of the function is known as 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 the same as every other one-to-one functions. This implies that the inverse of a one-to-one function will possess one domain for every range and pass the horizontal line test.
How do you find the inverse of a One-to-One Function?
Determining the inverse of a function is very easy. You just have to swap the x and y values. For instance, the inverse of the function f(x) = x + 5 is f-1(x) = x - 5.
As we reviewed before, the inverse of a one-to-one function reverses the function. Because 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 Questions
Consider 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 each of these functions:
1. Determine if the function is one-to-one.
2. Draw the function and its inverse.
3. Figure out the inverse of the function algebraically.
4. Indicate the domain and range of both the function and its inverse.
5. Apply the inverse to find the solution for x in each formula.
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