**OBJECTIVES**

1. Determining whether a function fits the definition of one-to-one

**Objective 1:** Determining Whether a Function fits the Definition of One-to-One

**Definition:** A function for which every element of the range of the function corresponds to exactly one element of the domain. A function is not one-to-one if two different elements in the domain correspond to the same element of the range.

**Theorem: Horizontal line-Test**

If every horizontal line intersects the graph of a function f(x) in at most one point, then f(x) is one-to-one.

**Example 1a:**

Does this table represent a one-to-one function? Why or why not?

Time (in hours) | Yeast Cells |
---|---|

0 | 56 |

2 | 15 |

4 | 65 |

6 | 60 |

8 | 28 |

**Solution 1a:**

According to the definition, each input correlates to one and only one output. So, this is a one-to-one function.

**Example 1b:**

Does this table represent a one-to-one function? Why or why not?

Time (in hours) | Yeast Cells |
---|---|

0 | 56 |

2 | 56 |

4 | 56 |

6 | 56 |

8 | 56 |

**Solution 1b:**

This table does not represent a one-to-one function because more than one input has the same output. Therefore, this is a function, but not a one-to-one function.

**Practice Problems:**

For Problems 1-5 assess each set of data and answer the question is this a one-to-one function?

1. {(1,4), (2,3), (-7,3)}

2. {(15,16), (34,27), (64,8)}

3. {(15,15), (10,15), (5,15)}

4. {(69,74), (54,75), (23,76)}

5. {(134,5), (123,7), (154,3)}

For Problems 6-10 graph each function and use horizontal-line test to determine whether f is one-to-one function:

6. f(x) = x

7. f(x) = x^{2}

8. f(x) = x^{(½)}

9. f(x) = x^{x}

10. f(x) = e^{x}