Rational and irrational functions
The total resistance of resistors placed in series in an electrical circuit and the length of the period of a pendulum are examples of rational and irrational functions. In this chapter you will review the properties of these types of functions, the arithmetic processes for rational and irrational expressions, and the techniques for solving rational and irrational equations.
Rational functions
If f(x) is equal to the ratio of two polynomial functions,g(x) and k(x) (i.e., 
a number of questions arise. For what values of x is f(x) defined (the domain)? For what values of x does f(x) = 0 (the zeroes)? What is the behavior of f(x) when the magnitude of the values of x become large (end behavior)? And what are the possible outcomes when f(x) is evaluated (the range)?
For what values of x is f(x) defined? Division by zero is undefined. Any value of x that makes the denominator equal to zero must be rejected.
For what values of x will f(x) = 0? A rational expression is equal to 0 when the numerator is 0 and the denominator is not.
There are a number of different possibilities for the end behavior of a rational function. For the level of the Algebra II curriculum, you will need to consider three cases: (1) the degree of the numerator is less than the degree of the denominator; (2) the degree of the numerator equals the degree of the denominator; and (3) the degree of the numerator is greater than the degree of the denominator. In each case, a simple explanation is to consider substituting a very large value for x in the function. Only the terms of largest degree in the numerator and denominator will matter, and the rest of the terms in the function can be ignored.
In general, if 
, the end behavior for f(x) is: (1) if m < n, the graph goes to 0; (2) if m = n, the graph goes to 
and(3) if m > n, the graph gets infinitely large.
What are the possible outcomes when f(x) is evaluated? This is a difficult question to answer because of the variety of rational functions you might encounter. You have seen from your study of inverse functions that the domain of the inverse function is the range of the original function. Computing the inverse of a rational function whose numerator and denominator are both linear expressions can be done. Other situations are not as straightforward, and the use of graphing technology is helpful for determining these ranges.
Given 
determine the following:
1. The domain of f(x)
2. The zeroes of f(x)
3. The end behavior of f(x)
Given 
determine the following:
4. The domain of g(x)
5. The zeroes of g(x)
6. The end behavior of g(x)
Given 
determine the following:
7. The domain of k(x)
8. The zeroes of k(x)
9. The end behavior of k(x)
10. k−1(x)
Multiplying and dividing rational expressions
Multiplying rational expressions involves eliminating common factors from the numerator and denominator of the constituent expressions, and then multiplying all remaining factors of the numerator and all remaining factors of the denominator.
When multiplying rational algebraic expressions, the process will be exactly the same. Factor the terms into their prime factors and then remove any factors common to the numerator and denominator.
Note: The domains of the original problem and the reduced problem must be the same. That is, the domain of the simplified answer is still 
The importance of looking for the domain of the original problem will become clearer when solving rational equations.
Since division is the inverse operation for multiplication, the process of dividing rational algebraic expressions is to change the division to multiplication, and change the divisor to its reciprocal. The problem then becomes a multiplication problem.
An important rule for you to remember is that expressions of the form 
are equal to −1. (If you are unsure of this, pick any two values for a and b and work out the problem.)
Adding and subtracting rational expressions
Just as adding and subtracting algebraic expressions require “like” terms, so do adding and subtracting rational expressions. While you can add 3x and 5x to make a single term, 8x, 3x + 5y is as simple as this particular expression can be. When adding 
you are adding the number of eighths. The purpose of getting a common denominator is so that you can add or subtract like terms.
A complex fraction is a fraction in which the numerator and/or the denominator contain fractions. To change a complex fraction to a simple fraction, multiply the numerator and denominator by the common denominator of the component fractions.
Solving rational equations
The basic principle for solving rational equations is to multiply by a common denominator to remove the fraction from the problem. Pay careful attention to the domain of the rational expressions and compare your solutions against these values.
Solve the following.
1. 
2. 
3. 
4. 
5. 
6. 
7. Capacitance is the measure of a capacitor’s ability to hold a charge. If a set of capacitors are put in a circuit in series, the total equivalent capacitance, Ceq, is given by the formula 
for as many capacitors as are in the circuit. If three capacitors with ratings c farads, c + 20 farads, and 3c farads are placed in a series, the net capacitance is 20 farads. Find the value of each capacitor.
Irrational functions
As discussed in Chapter 5, functions of the form 
are the inverse of the functions g(x) = xn. When n is a positive even integer, the domain of f(x) is x ≥ 0, and the range of f(x) is y ≥ 0. When n is a positive odd integer, the domain and range of f(x) are the set of real numbers.
Given 
determine the following:
1. The domain of f(x)
2. The range of f(x)
Given 
determine the following:
3. The domain of g(x)
4. The range of g(x)
Given 
determine the following:
5. The domain of k(x)
6. The range of k(x)
Given 
determine the following:
7. The domain of m(x)
8. The range of m(x)
Simplifying irrational expressions
Some basic rules of working with irrational expressions are perfect powers larger than 1 cannot be kept within the radical; radicands should not contain fractions; and it is common practice to rationalize the denominators of fractions.
Simply the expressions in questions 1–4.
1. 
2. 
3. 
4. 
Rationalize the denominator for each fraction given in questions 5–7.
5. 
6. 
7. 
Solving irrational equations
The basic format for solving irrational equations is to isolate the radical expression on one side of the equation, and then raise both sides of the resulting equation to an appropriate power to remove the radical.
