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 corresponds to only one 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 noted as the range of the function.

Let's examine the pictures below:

Source

For f(x), any value in the left circle corresponds to a unique value in the right circle. In the same manner, any value in the right circle correlates to a unique value in the left circle. In mathematical terms, this signifies every domain holds a unique range, and every range owns a unique domain. Thus, this is a representation of a one-to-one function.

Here are some additional examples of one-to-one functions:

• f(x) = x + 1

• f(x) = 2x

Now let's look at the second example, which shows the values for g(x).

Be aware of the fact that the inputs in the left circle (domain) do not hold unique outputs in the right circle (range). For instance, the inputs -2 and 2 have equal output, i.e., 4. In conjunction, the inputs -4 and 4 have equal output, i.e., 16. We can comprehend that there are identical 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:

• f(x) = x^2

• f(x)=(x+2)^2

What are the characteristics of One to One Functions?

One-to-one functions have the following characteristics:

• The function holds an inverse.

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

• They pass the horizontal line test.

• The graph of a function and its inverse are identical regarding the line y = x.

How to Graph a One to One Function

To graph a one-to-one function, you are required to find the domain and range for the function. Let's examine an easy representation of a function f(x) = x + 1.

As soon as you have the domain and the range for the function, you ought to graph the domain values on the X-axis and range values on the Y-axis.

How can you tell if 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 chart the graph of a function, draw horizontal lines over the graph. If a horizontal line passes 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 reason that all linear functions are one-to-one functions. Remember that we do not apply the vertical line test for one-to-one functions.

Let's study the graph for f(x) = x + 1. Once you graph the values to x-coordinates and y-coordinates, you need to consider if a horizontal line intersects the graph at more than one point. 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 multiple times. Let's look at the figure 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 multiple horizontal lines. Case in point, for either domains -1 and 1, the range is 1. Additionally, for each -2 and 2, the range is 4. This means that f(x) = x^2 is not a one-to-one function.

What is the inverse of a One-to-One Function?

As 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 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, in other words, 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 have one domain for every range and pass the horizontal line test.

How do you determine the inverse of a One-to-One Function?

Finding the inverse of a function is not difficult. 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.

Source

As we reviewed previously, the inverse of a one-to-one function reverses the function. Considering the original output value showed us we needed to add 5 to each input value, the new output value will require us to delete 5 from each input value.

One to One Function Practice Questions

Consider the following 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 whether or not the function is one-to-one.

2. Draw the function and its inverse.

3. Figure out the inverse of the function mathematically.

4. Indicate the domain and range of each function and its inverse.

5. Apply the inverse to determine the value for x in each equation.

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