# Functions, Graphs, and Limits

## Introduction to Functions, Graphs, And Limits - At A Glance:

There are three steps to solving a math problem.

• Figure out what the problem is asking.
• Solve the problem.

### Sample Problem

Graph the function

• Figure out what the problem is asking.

This problem is asking for a lot. While it may seem like it's asking for a picture,
the picture needs to show any holes, vertical asymptotes, horizontal asymptotes, and x or y intercepts of the function.

• Solve the problem.

First we need to factor the function:

If we simplify, we find

Since the term x is removed from the denominator after simplifying, the function has a hole at x = 0. The full coordinates of the hole are (0,-\frac49). Since the expressions (x - 3) and (x + 3) are still in the denominator after simplifying, there will be vertical asymptotes at 3 and -3.

Since the degree of the numerator is less than the degree of the denominator, the function will have a horizontal asymptote at 0.

In terms of the picture, here's what we have now:

Now we need to figure out the sign of the function f using a number line:

When x < -3, both (x-3) and (x + 3) are negative, therefore f is positive.

When -3 < x < 3 we know (x - 3) is negative and (x + 3) is positive, therefore f is negative.

When 3 < x both (x - 3) and (x + 3) are positive, therefore f is positive.

Now we can fill in the number line:

Now we have enough information to draw the graph. Since f can't change sign on the interval (-∞,-3) or on the interval (3,∞), it must look like this: