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 in which each input correlates to just one output. That is to say, for every x, there is just 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 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. In conjunction, any value on the right side corresponds to a unique value on the left. In mathematical words, this signifies every domain holds a unique range, and every range has a unique domain. Hence, this is a representation 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 example, 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 example, the inputs -2 and 2 have identical output, i.e., 4. In the same manner, the inputs -4 and 4 have equal output, i.e., 16. We can comprehend that there are identical Y values for multiple X values. Thus, this is not a one-to-one function.
Here are different 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 properties:
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The function holds 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 the same with respect to 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 determine the domain and range for the function. Let's examine a straight-forward example of a function f(x) = x + 1.
Once you have the domain and the range for the function, you ought 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 indicate if a function is one-to-one, we can apply the horizontal line test. As soon as you plot the graph of a function, draw horizontal lines over the graph. If a horizontal line intersects 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 point, we can also deduct all linear functions are one-to-one functions. Remember that we do not leverage the vertical line test for one-to-one functions.
Let's examine the graph for f(x) = x + 1. Immediately after you plot the values of x-coordinates and y-coordinates, you ought to consider whether 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 contrary, if the function is not a one-to-one function, it will intersect the same horizontal line more than one time. 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 meets numerous horizontal lines. Case in point, for either domains -1 and 1, the range is 1. In the same manner, for each -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 just 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 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 as a means of getting the output, i.e., y. The inverse of this function will deduct 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 the same as all other one-to-one functions. This signifies 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 find the inverse of a One-to-One Function?
Figuring out the inverse of a function is very easy. You simply need to switch the x and y values. For example, the inverse of the function f(x) = x + 5 is f-1(x) = x - 5.
Just like we reviewed before, the inverse of a one-to-one function undoes the function. Because 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
Examine 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. Figure out if the function is one-to-one.
2. Graph the function and its inverse.
3. Find the inverse of the function algebraically.
4. State the domain and range of each function and its inverse.
5. Apply the inverse to find the solution for x in each calculation.
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