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Functions, Graphs, and Limits

Functions, Graphs, and Limits

How to Solve a Math Problem

There are three steps to solving a math problem.

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

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 . 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:

  • Check the answer.

For a problem like this, it's probably necessary to double check the solution. Look over the graph to make sure everything is labeled. If necessary, make sure the asymptotes are correct and known values of the function are labeled.

Use a graphing calculator to check that any graph makes sense.

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