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 corresponds to just one output. That is to say, for each x, there is just 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 noted as the domain of the function, and the output value is the range of the function.
Let's look at the examples below:
For f(x), each value in the left circle corresponds to a unique value in the right circle. In the same manner, every value in the right circle corresponds to a unique value on the left side. In mathematical terms, this means that every domain holds a unique range, and every range holds 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 study the second example, which exhibits the values for g(x).
Be aware of the fact that the inputs in the left circle (domain) do not own unique outputs in the right circle (range). For example, the inputs -2 and 2 have equal output, in other words, 4. Similarly, the inputs -4 and 4 have identical output, i.e., 16. We can comprehend that there are equivalent Y values for many X values. Therefore, 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 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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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
When trying to graph a one-to-one function, you will have to find the domain and range for the function. Let's examine a straight-forward example of a function f(x) = x + 1.
Once you know the domain and the range for the function, you have to graph the domain values on the X-axis and range values on the Y-axis.
How can you determine if a Function is One to One?
To test whether a function is one-to-one, we can apply the horizontal line test. As soon as you plot the graph of a function, trace horizontal lines over the graph. If a horizontal line moves through the graph of the function at more than one point, then the function is not one-to-one.
Since the graph of every linear function is a straight line, and a horizontal line will not intersect the graph at more than one place, we can also deduct all linear functions are one-to-one functions. Remember 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 of x-coordinates and y-coordinates, you have to consider if a horizontal line intersects the graph at more than one spot. In this example, the graph does not intersect any horizontal line more than once. This indicates 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 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 crosses various horizontal lines. Case in point, 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?
As 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 undoes 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 subtract 1 from each value of y.
The inverse of the function is known as f−1.
What are the properties of the inverse of a One to One Function?
The characteristics of an inverse one-to-one function are identical to every other one-to-one functions. This signifies that the opposite of a one-to-one function will have one domain for each 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 simple. You simply have to switch the x and y values. For instance, the inverse of the function f(x) = x + 5 is f-1(x) = x - 5.
As we learned earlier, the inverse of a one-to-one function reverses the function. Considering 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 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 every function:
1. Figure out if the function is one-to-one.
2. Graph the function and its inverse.
3. Figure out the inverse of the function algebraically.
4. State the domain and range of each function and its inverse.
5. Employ the inverse to solve for x in each equation.
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