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

Let's look at the pictures below:

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

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

• f(x) = x + 1

• f(x) = 2x

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

Notice that the inputs in the left circle (domain) do not have unique outputs in the right circle (range). For instance, the inputs -2 and 2 have equal output, i.e., 4. Similarly, the inputs -4 and 4 have the same output, i.e., 16. We can comprehend that there are equivalent Y values for numerous X values. Hence, this is not a one-to-one function.

Here are different 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 qualities:

• 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 the same with respect to the line y = x.

How to Graph a One to One Function

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

Once you possess 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 evaluate if a Function is One to One?

To test whether or not a function is one-to-one, we can apply the horizontal line test. Once you chart the graph of a function, trace horizontal lines over the graph. In the event that 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 point, we can also deduct 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 study the graph for f(x) = x + 1. Once you chart the values of x-coordinates and y-coordinates, you have to examine 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 means 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 more than one time. Let's look at the diagram 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 meets numerous horizontal lines. Case in point, for either domains -1 and 1, the range is 1. In the same manner, 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 opposite 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 opposite of the function essentially undoes the function.

For example, in the event of f(x) = x + 1, we add 1 to each value of x as a means 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 known as f−1.

What are the qualities of the inverse of a One to One Function?

The characteristics of an inverse one-to-one function are identical to every other one-to-one functions. This means that the opposite 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?

Figuring out the inverse of a function is very easy. You simply need 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

Just like we discussed previously, the inverse of a one-to-one function reverses the function. Since the original output value required us to add 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

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. Figure out whether the function is one-to-one.

2. Plot the function and its inverse.

3. Determine the inverse of the function numerically.

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

5. Use the inverse to solve for x in each equation.

Grade Potential Can Help You Learn You Functions

If you are having problems trying to understand one-to-one functions or similar concepts, Grade Potential can connect you with a one on one tutor who can help. Our Santa Barbara math tutors are skilled professionals who support students just like you improve their understanding of these concepts.

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