1. Rational Functions: Graphs, Applications, and Models
Chapter 3.5
Rational Functions: Graphs, Applications, and Models

3. The Reciprocal Function
The simplest rational function with a variable denominator is called the reciprocal function, defined by

4. The Reciprocal Function
The domain of this function is the set of all real numbers except 0. The number 0 cannot be used as a value of x, but it is helpful to find values of f(x) for some values of x close to 0.

5. The Reciprocal Function
The domain of this function is the set of all real numbers except 0. The number 0 cannot be used as a value of x, but it is helpful to find values of f(x) for some values of x close to 0.

6. The Reciprocal Function
We use the table feature of a graphing calculator to do this. The tables in Figure 37 suggest that |f(x)| gets larger and larger as x gets closer and closer to 0, which is written in symbols as

7. The Reciprocal Function

8. The Reciprocal Function

9. The Reciprocal Function

10. The Reciprocal Function

11. The Reciprocal Function

13. y x

14. y x

19. y x

26. Steps for Graphing Rational Functions
A comprehensive graph of a rational function exhibits these features:
all x- and y-intercepts;
all asymptotes: vertical, horizontal or oblique.
the point at which the graph intersects its non-vertical asymptote (if there is any such point.
enough of the graph to exhibit the correct end behavior.

28. x

31. x

33. x

36. x

38. x