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What is a One-to-One Function?

What is a one-to-one function?

Summary

  1. In the mapping diagram, 'D' stands for 'domain' and 'R' stands for 'range'
  2. Each element in the range of 'A' has one arrow pointing to it, so it is a 1-1 function
  3. '1-1' is another way of writing 'one-to-one'
  4. In diagram 'B', 12 has two arrows pointing to it, so it is NOT 1-1
  5. The graph of 'C' passes both the vertical AND horizontal line tests, so it is 1-1
  6. The graph of 'D' passes the vertical line test but NOT the horizontal, so it is NOT 1-1

Notes

    1. A relation is just a set of ordered pairs
    2. The domain is the set of all the inputs, or x-values, in the relation
    3. The range is the set of all the outputs, or y-values, in the relation
    1. To be a function, each element of the domain can only be paired with ONE element of the range
    2. In the first mapping diagram, there is only one arrow coming from each domain value
    3. That means each domain value is only paired with one range value, so the relation IS a function
    4. In the second mapping diagram, there are TWO arrows coming from 8
    5. That means 8 is paired with TWO range values, so the second relation is NOT a function
    1. '1-1' is another way of writing 'one-to-one'
    1. This sounds like the reverse of the definition of a function!
    2. But a one-to-one function must also follow the definition for a function!
    3. So it's domain values can still only be paired with one range value
    4. This basically means that each value in the domain gets its own value in the range, and doesn't have to share with any others!
    1. Each element in the domain of 'A' has one arrow coming from it in the mapping diagram, so it is a function
    1. Each element in the range only has one arrow pointing to it
    2. This means that each range value is paired with EXACTLY ONE domain value
    3. So this function is one-to-one!
    1. Each element in the domain of 'B' has one arrow coming from it in the mapping diagram, so it is a function
    1. 12 has TWO arrows pointing to it
    2. That means it is paired with TWO elements of the domain
    3. So even though 'B' IS a function, it is NOT a 1-1 function
    4. '1-1' is another way of writing 'one-to-one'
    1. '1-1' is another way of writing 'one-to-one'
    1. We can tell if a graph represents a function just by looking at it!
    1. If a vertical line goes through the graph more than once, that means there is more than one y-value for that x-value
    2. And remember, each x-value can only be paired with ONE y-value if it is a function
    3. So if a graph fails the vertical line test, it is NOT a function
    1. The horizontal line test is just like the vertical line test, except we draw horizontal lines through the graph instead
    2. If a horizontal line only goes through the graph once, that means there is only one x-value for that y-value
    3. And remember, to have a one-to-one function each y-value can only be paired with one x-value
    1. Since Graph C passes both the vertical AND horizontal line tests, it IS a 1-1 function!
    1. We'll use the vertical line test first to make sure the graph represents a function
    2. Any vertical line you draw will cross the graph only once, so Graph D passes the vertical line test and IS a function
    3. Then we'll use the horizontal line test to see if the function is one-to-one
    4. We can draw a horizontal line that crosses the graph in two places
    5. So Graph D fails the horizontal line test and is NOT one-to-one
    1. Since it failed the horizontal line test, we can tell that Graph D does not represent a 1-1 function