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 only one output. So, 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 the domain of the function, and the output value is known as the range of the function.

Let's study the examples below:

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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 on the left side. In mathematical terms, this signifies every domain owns a unique range, and every range owns a unique domain. Hence, this is an example 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 examine the second example, which shows 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). Case in point, the inputs -2 and 2 have equal output, in other words, 4. Similarly, the inputs -4 and 4 have the same output, i.e., 16. We can discern that there are matching Y values for numerous X values. Therefore, this is not a one-to-one function.

Here are some other examples 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 qualities:

• The function has an inverse.

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

• The function passes the horizontal line test.

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

How to Graph a One to One Function

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

Immediately after you possess the domain and the range for the function, you need to graph the domain values on the X-axis and range values on the Y-axis.

How can you evaluate whether or not a Function is One to One?

To indicate if a function is one-to-one, we can use the horizontal line test. Once 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 place, 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 reason that all linear functions are one-to-one functions. Don’t forget that we do not apply the vertical line test for one-to-one functions.

Let's look at the graph for f(x) = x + 1. Once you graph the values to x-coordinates and y-coordinates, you ought to review whether 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.

Subsequently, 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 instance, the graph crosses multiple horizontal lines. For instance, for both domains -1 and 1, the range is 1. Additionally, for both -2 and 2, the range is 4. This implies 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 undoes the function.

For example, in the event of f(x) = x + 1, we add 1 to each value of x in order to get the output, or y. The inverse 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 the same as any other one-to-one functions. This means that the reverse of a one-to-one function will hold 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 simple. 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.

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Just like we learned previously, the inverse of a one-to-one function reverses the function. Because the original output value required adding 5 to each input value, the new output value will require us to delete 5 from each input value.

One to One Function Practice Examples

Contemplate these 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 each of these functions:

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

2. Graph the function and its inverse.

3. Figure out the inverse of the function mathematically.

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

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

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