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 where each input correlates to only one output. In other words, for each x, there is just 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 noted as the domain of the function, and the output value is noted as the range of the function.
Let's examine the examples below:
For f(x), any value in the left circle corresponds to a unique value in the right circle. In conjunction, any value on the right side corresponds to a unique value in the left circle. In mathematical words, 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 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 example, which displays the values for g(x).
Pay attention to 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. In the same manner, the inputs -4 and 4 have the same output, i.e., 16. We can discern that there are matching Y values for multiple X values. Hence, this is not a one-to-one function.
Here are some other 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 characteristics 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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They pass 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
To graph a one-to-one function, you will need to determine the domain and range for the function. Let's study a straight-forward representation of a function f(x) = x + 1.
Immediately after you possess the domain and the range for the function, you need to plot the domain values on the X-axis and range values on the Y-axis.
How can you determine whether or not a Function is One to One?
To prove if a function is one-to-one, we can use the horizontal line test. Immediately after you graph the graph of a function, trace horizontal lines over the graph. In the event that 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 place, we can also conclude 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 study the graph for f(x) = x + 1. Once you chart the values to x-coordinates and y-coordinates, you need to consider 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 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 once. Let's look at 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 intersects multiple 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 signifies 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 a single 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 example, in the example 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 denoted as f−1.
What are the characteristics of the inverse of a One to One Function?
The properties of an inverse one-to-one function are identical to every other one-to-one functions. This implies that the inverse 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 very easy. You just 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.
Just like we discussed before, 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
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 each of these functions:
1. Determine if the function is one-to-one.
2. Draw the function and its inverse.
3. Find the inverse of the function mathematically.
4. Specify the domain and range of each function and its inverse.
5. Use the inverse to solve for x in each equation.
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