One-to-One Functions

A function is one-to-one if no two distinct values in the domain correspond to the same value in the range.

For example, the function f (x) = x2 is not one-to-one since

f (- 3) = 9 and f (3) = 9.

In other words, two distinct domain values, - 3 and 3, correspond to the same range value, 9.

One-to-One Functions

There is a horizontal line test associated with the definition.

If it there exists a horizontal line that touches the graph of a function in more than one place, the function is not one-to-one. Otherwise, the function is one-to-one.

One-to-One Functions

To see why the horizontal line test works, look again at the function, f (x) = x2. Earlier, using the definition, it was found not to be one-to-one.

Now look at its graph.

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-4 -3 -2 -1 1 2 3 4

(3, 9)(- 3, 9)You can see that there exists a horizontal line that touches the graph at more than one place, (- 3, 9) and (3, 9).

It only takes one horizontal line touching the graph at more than one place to conclude the function is not one-to-one.

One-to-One Functions

Try: Graph and use the horizontal line test to determinewhether each function is one-to-one.

(a) f (x) = x3 – x – 2. It is not one-to-one.

One-to-One Functions

Try: Graph and use the horizontal line test to determinewhether each function is one-to-one.

(b) f (x) = x3 + x – 2. It is one-to-one.

One-to-One Functions