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 a single output. In other words, for every 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 known as the domain of the function, and the output value is known as the range of the function.

Let's look at the examples below:

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For f(x), every value in the left circle correlates to a unique value in the right circle. Similarly, each value on the right side correlates to a unique value on the left. In mathematical words, this implies 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 more representations of one-to-one functions:

• f(x) = x + 1

• f(x) = 2x

Now let's examine the second picture, 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, 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 many X values. Thus, this is not a one-to-one function.

Here are some other representations of non one-to-one functions:

• f(x) = x^2

• f(x)=(x+2)^2

What are the characteristics of One to One Functions?

One-to-one functions have these properties:

• The function has an inverse.

• The graph of the function is a line that does not intersect itself.

• It passes the horizontal line test.

• The graph of a function and its inverse are equivalent with respect to the line y = x.

How to Graph a One to One Function

In order to graph a one-to-one function, you will have to figure out the domain and range for the function. Let's study a simple example of a function f(x) = x + 1.

Immediately after you have 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 evaluate whether a Function is One to One?

To indicate whether or not a function is one-to-one, we can use 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 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 will not intersect the graph at more than one point, we can also conclude 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. Immediately after you plot the values to x-coordinates and y-coordinates, you have to review if a horizontal line intersects the graph at more than one point. In this case, 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 multiple times. 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 case, the graph intersects various horizontal lines. For example, for each domains -1 and 1, the range is 1. Additionally, 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?

Since a one-to-one function has a single 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 undoes the function.

For Instance, in the example of f(x) = x + 1, we add 1 to each value of x for the purpose 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 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 identical to all other one-to-one functions. This implies that the opposite of a one-to-one function will possess 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 very easy. 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.

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Just like we reviewed before, the inverse of a one-to-one function reverses the function. Since the original output value required adding 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

Examine the subsequent functions:

• f(x) = x + 1

• f(x) = 2x

• f(x) = x2

• f(x) = 3x - 2

• f(x) = |x|

• g(x) = 2x + 1

• h(x) = x/2 - 1

• j(x) = √x

• k(x) = (x + 2)/(x - 2)

• l(x) = 3√x

• m(x) = 5 - x

For every function:

1. Identify if the function is one-to-one.

2. Plot the function and its inverse.

3. Figure out the inverse of the function numerically.

4. Specify the domain and range of both the function and its inverse.

5. Apply the inverse to find the solution for x in each calculation.

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