Chapter 8
Rational Expressions and Functions
Split a sixty dollar bill among the people at your table: each share is sixty divided by however many turned up. Replace that count with a letter and a single line covers every table at once. Rational expressions are the tool for writing divisions like that, where the amount underneath changes as the letter does. That flexibility comes with a catch. A bottom that changes can pass through zero, and division has never allowed that. One question then follows the chapter the whole way: what happens at an input the expression cannot accept?
What You'll Explore
5 lessons.
- Simplifying Rational Expressions
A rational expression can often be written in a shorter form that looks easier to work with. But is the short form really the same expression, or only a close relative? You will look at which pieces are allowed to cancel, and at a question that has to be settled before any cancelling starts.
- Multiplying and Dividing Rational Expressions
Multiplying two ordinary fractions takes a single line, and rational expressions borrow the same move. What is new is the order the work has to happen in, along with a complication that division introduces and then hides, leaving no sign of itself in the final answer.
- Adding and Subtracting Rational Expressions
Two rational expressions with different denominators cannot simply be pushed together, and building common ground between them takes more care than it does with plain numbers. Subtraction is where most of the errors in this chapter live, and it is worth finding out exactly where they come from.
- Solving Rational Equations
Once rational expressions can be combined, an equation made of them should be no harder than any other. Mostly it is not. But this kind of equation can hand back a perfectly clean answer that the original refuses to accept, and telling the difference becomes part of the work.
- Graphs of Rational Functions
Every function in this chapter has a graph, and it will not look like the smooth curves you are used to. Near an input the expression cannot accept, the picture does something dramatic, and not always the same thing twice. You will find out what the graph does there, and where it heads far from the middle.