Lesson 18

Decide if each of these equations is true or false for \(x\) values that do not result in a denominator of 0. Be prepared to explain your reasoning.

1. \(\displaystyle{\frac{x+7}{x}=1+\frac{7}{x}}\)
2. \(\displaystyle{\frac{x}{x+7} =1+\frac{x}{7}}\)

Let \(c\) be the function that gives the average cost per book \(c(x)\), in dollars, when using an online store to print \(x\) copies of a self-published paperback book. Here is a graph of \(c(x)= \tfrac{120+4x}{x}.\)

1. What is the approximate cost per book when 50 books are printed? 100 books?
2. The author plans to charge $8 per book. About how many should be printed to make a profit?
3. What is the value of \(c(x)\) when \(x=\frac{1}{2}\)? How does this relate to the context?
4. What does the end behavior of the function say about the context?

Here are four graphs of rational functions.

1. Match each function with its graphical representation.
   1. \(a(x)=\frac{4}{x}-1\)
   2. \(b(x)=\frac{1}{x}-4\)
   3. \(c(x)=\frac{1+4x}{x}\)
   4. \(d(x)=\frac{x+4}{x}\)
   5. \(e(x)=\frac{1-4x}{x}\)
   6. \(f(x)=\frac{4-x}{x}\)
   7. \(g(x)=1+\frac{4}{x}\)
   8. \(h(x)=\frac{1}{x}+4\)
2. Where do you see the horizontal asymptote of the graph in the expressions for the functions?

Consider the rational function \(f(x) = \frac{3x+1}{x}\). Written this way, we can tell that the graph of the function has a vertical asymptote at \(x=0\) by reading the denominator and identifying the value that would cause division by zero. But what can we tell about the value of \(f(x)\) for values of \(x\) far away from the vertical asymptote?

Written this way, it’s easier to see that as \(x\) gets larger and larger in either the positive or negative direction, the \(\frac{1}{x}\) term will get closer and closer to 0. Because of this, we can say that the value of the function will get closer and closer to 3.

More generally, if a rational function \(g(x) = \frac{a(x)}{b(x)}\) can be rewritten as \(g(x) = c + \frac{r(x)}{b(x)}\), where \(c\) is a constant, and \(r(x)\) and \(b(x)\) are polynomial expressions where \(\frac{r(x)}{b(x)}\) gets closer and closer to zero as \(x\) gets larger and larger in both the positive and negative directions, then \(g(x)\) will get closer and closer to \(c\).

Rational functions of this type have a horizontal asymptote at the constant value. The line \(y=c\)  is a horizontal asymptote for \(f\) if \(f(x)\) gets closer and closer to \(c\) as the magnitude of \(x\) increases.