Example:

Evaluate when and

Solution:

Replace x with -3 in each function | |

is undefined.

Example:

Evaluate when and

Solution:

Replace x with -3 in each function | |

is undefined.

Example: Find the difference quotient for

The Difference Quotient:

Solution:

Write the difference quotient for the given function | |

Apply the exponent and use the distributive property | |

Multiply | |

Example: Find the difference quotient for

The Difference Quotient:

Solution:

Write the difference quotient for the given function | |

Use the distributive property | |

Simplify the complex fraction by multiplying the numerator and denominator by the common denominator | |

Example: Find the difference quotient for

The Difference Quotient:

Solution:

Write the difference quotient for the given function | |

Simplify the complex fraction by multiplying the numerator and denominator by the common denominator | |

Distribute the common denominator to each fraction in the numerator. | |

The difference quotient for is

Example: For the function below. Graph using transformations. Find the y-intercept. State the horizontal asymptote and the domain and range.

First we must examine the base function

Graph using plotting points. We can use the standard set of x-values to find ordered pairs.

x | y |
---|---|

-2 | 2^(-2)=1/4 |

-1 | 2^(-1)=1/2 |

0 | 2^0=1 |

1 | 2^1=2 |

2 | 2^2=4 |

The graph below shows the points plotted and the line that connects them. This graph has a horizontal asymptote at y=0. The domain is and the range is

Analyze the transformations.

The +2 in the exponent shifts the graph left 2 units.

The – in the front of the base reflects the graph over the x-axis.

The +2 next to the base shifts the graph and the horizontal asymptote up two units.

You can see the graph after the transformations.

The horizontal asymptote is y=2. The domain is and the range is

**To find the y-intercept we let x=0.**

Thus the y-intercept is (0,-2)

**The Distance Formula**

Suppose A is and B is

The **distance** between points A and B is given by the following formula.

**Example:** Find the distance between points A and B.

Point A is and point B is