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 a single output. So, for every x, there is a single 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 the domain of the function, and the output value is noted as the range of the function.

Let's examine the pictures below:

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For f(x), every value in the left circle corresponds to a unique value in the right circle. Similarly, every value on the right side correlates to a unique value in the left circle. In mathematical words, this signifies every domain owns a unique range, and every range has a unique domain. Hence, this is an example of a one-to-one function.

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

• f(x) = x + 1

• f(x) = 2x

Now let's study the second image, which displays the values for g(x).

Notice 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 the same output, in other words, 4. In the same manner, the inputs -4 and 4 have equal 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 different examples of non one-to-one functions:

• f(x) = x^2

• f(x)=(x+2)^2

What are the properties of One to One Functions?

One-to-one functions have these characteristics:

• The function owns 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

When trying 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 know 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 determine if a Function is One to One?

To test whether a function is one-to-one, we can use the horizontal line test. As soon as you graph 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.

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 deduct 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 examine the graph for f(x) = x + 1. As soon as you plot the values of x-coordinates and y-coordinates, you ought to examine whether 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 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 more than one time. Let's examine 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 crosses various 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 signifies 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 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 basically undoes the function.

Case in point, in the example 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 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 the same as any other one-to-one functions. This means that the inverse of a one-to-one function will possess one domain for each range and pass the horizontal line test.

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

Finding the inverse of a function is not difficult. You simply 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

Considering what we learned previously, 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 subtract 5 from each input value.

One to One Function Practice Questions

Consider 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 any of these functions:

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

2. Draw the function and its inverse.

3. Figure out the inverse of the function mathematically.

4. Specify 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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