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 Depdendent Variable

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 Dependent Variable

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Horizontal Line Test

A function where each y-value (output) corresponds to exactly one x-value (input) is called a one-to-one function. A one-to-one function has an inverse function.

To determine if a function is one-to-one, we can use the horizontal line test.

Note:

Another way to think about a function that is one-to-one is the following: Two different input values always result in 2 different output values.

Procedure â€” To Determine if a Graph Represents a One-To-One Function (Horizontal Line Test)

If you can draw a horizontal line anywhere through the graph of a function and it intersects the graph at most once, then the graph represents a one-to-one function.

Example 1

Given the graph of the function: f(x) = -2x + 4

a. Is f(x) a one-to-one function?

b. Does f(x) have an inverse?

Solution

a. To determine if the function is one-to-one, use the horizontal line test. Any horizontal line intersects the graph at most once. Thus, each output corresponds to at most one input.

Therefore, f(x) = -2x + 4 is a one-to-one function.

b. Since the function is one-to-one, f(x) has an inverse.

Note:

Any linear function is one-to-one unless it has the form f(x) =  c, where c is a constant.

The graph of f(x) = c is a horizontal line and so it does not pass the horizontal line test.

Example 2

Given the graph of the function: f(x) = -1(x - 3)2 + 4

a. Is f(x) a one-to-one function?

b. Does f(x) have an inverse?

Solution

a. To determine if the function is one-to-one, use the horizontal line test. Since a horizontal line may intersect the graph more than once, some outputs correspond to more than one input.

For example, the y-value 3 corresponds to two x-values (2 and 4). Therefore, this function is not one to one.

b. Since the function is not one-to-one, it does not have an inverse that is a function.