1. Chapter 6 Rational Expressions, Functions, and Equations

2. § 6.1
Rational Expressions and Functions: Multiplying and Dividing

3. Rational Expressions
A rational expression consists of a polynomial divided by a nonzero polynomial (denominator cannot be equal to 0).
A rational function is a function defined by a formula that is a rational expression. For example, the following is a rational function:
Blitzer, Intermediate Algebra, 5e – Slide #3 Section 6.1

4. Rational Expressions The rational function
EXAMPLE
The rational function
models the cost, f (x) in millions of dollars, to inoculate x% of the population against a particular strain of flu. The graph of the rational function is shown. Use the function’s equation to solve the following problem.
Find and interpret f (60). Identify your solution as a point on the graph.
p 393
Blitzer, Intermediate Algebra, 5e – Slide #4 Section 6.1

5. Rational Expressions CONTINUED
Blitzer, Intermediate Algebra, 5e – Slide #5 Section 6.1

6. Rational Expressions
CONTINUED
SOLUTION
We use substitution to evaluate a rational function, just as we did to evaluate other functions in Chapter 2.
This is the given rational function.
Replace each occurrence of x with 60.
Perform the indicated operations.
Blitzer, Intermediate Algebra, 5e – Slide #6 Section 6.1

7. Rational Expressions
CONTINUED
Thus, f (60) = This means that the cost to inoculate 60% of the population against a particular strain of the flu is $195 million. The figure below illustrates the solution by the point (60,195) on the graph of the rational function.
(60,195)
Blitzer, Intermediate Algebra, 5e – Slide #7 Section 6.1

8. Rational Expressions - Domain
EXAMPLE
Find the domain of f if
SOLUTION
The domain of f is the set of all real numbers except those for which the denominator is zero. We can identify such numbers by setting the denominator equal to zero and solving for x.
Set the denominator equal to 0.
Factor.
p 393
Blitzer, Intermediate Algebra, 5e – Slide #8 Section 6.1

9. Rational Expressions - Domain
CONTINUED or
Set each factor equal to 0.
Solve the resulting equations.
Because 4 and 9 make the denominator zero, these are the values to exclude. Thus, or
Blitzer, Intermediate Algebra, 5e – Slide #9 Section 6.1

10. Rational Expressions - Domain
CONTINUED
In this example, we excluded 4 and 9 from the domain. Unlike the graph of a polynomial which is continuous, this graph has two breaks in it – one at each of the excluded values. Since x cannot be 4 or 9, there is not a function value corresponding to either of those x values. At 4 and at 9, there will be dashed vertical lines called vertical asymptotes.
The graph of the function will approach these vertical lines on each side as the x values draw closer and closer to each of them, but will not touch (cross) the vertical lines. The lines x = 4 and x = 9 each represent vertical asymptotes for this particular function.
p 395
Blitzer, Intermediate Algebra, 5e – Slide #10 Section 6.1

11. Horizontal Asymptotes Simplifying Rational Expressions
Vertical Asymptotes
A vertical line that the graph of a function approaches, but does not touch.
Horizontal Asymptotes
A horizontal line that the graph of a function approaches as x gets very large or very small. The graph of a function may touch/cross its horizontal asymptote.
Simplifying Rational Expressions
1) Factor the numerator and the denominator completely.
2) Divide both the numerator and the denominator by any common factors.
p 395
Blitzer, Intermediate Algebra, 5e – Slide #11 Section 6.1

12. Rational Expressions - Domain
Check Point 2
Find the domain of f if
SOLUTION
Set the denominator equal to 0.
Factor. or
Set each factor equal to 0.
Solve the resulting equations. or
p 394
Blitzer, Intermediate Algebra, 5e – Slide #12 Section 6.1

13. Simplifying Rational Expressions
EXAMPLE
Simplify:
SOLUTION
Factor the numerator and denominator.
Divide out the common factor, x + 1.
Simplify.
Blitzer, Intermediate Algebra, 5e – Slide #13 Section 6.1

14. Simplifying Rational Expressions
Check Point 3
Simplify:
SOLUTION
Factor the numerator and denominator.
Divide out the common factor, x + 1.
Simplify.
p 397
Blitzer, Intermediate Algebra, 5e – Slide #14 Section 6.1

15. Simplifying Rational Expressions
EXAMPLE
Simplify:
SOLUTION
Factor the numerator and denominator.
Rewrite 3 – x as (-1)(-3 + x).
Rewrite -3 + x as x – 3.
Blitzer, Intermediate Algebra, 5e – Slide #15 Section 6.1

16. Simplifying Rational Expressions
CONTINUED
Divide out the common factor, x – 3.
Simplify.
Do Check 4a and 4b on page 397
Blitzer, Intermediate Algebra, 5e – Slide #16 Section 6.1

17. Multiplying Rational Expressions
1) Factor all numerators and denominators completely.
2) Divide numerators and denominators by common factors.
3) Multiply the remaining factors in the numerators and multiply the remaining factors in the denominators.
p 398
Blitzer, Intermediate Algebra, 5e – Slide #17 Section 6.1

18. Multiplying Rational Expressions
EXAMPLE
Multiply:
SOLUTION
This is the original expression.
Factor the numerators and denominators completely.
Divide numerators and denominators by common factors.
Blitzer, Intermediate Algebra, 5e – Slide #18 Section 6.1

19. Multiplying Rational Expressions
CONTINUED
Multiply the remaining factors in the numerators and in the denominators.
Note that when simplifying rational expressions or multiplying rational expressions, we just used factoring.
With one additional step that is provided in the following
Definition for Division, division of rational expressions promises to be just as straightforward.
Blitzer, Intermediate Algebra, 5e – Slide #19 Section 6.1

20. Multiplying Rational Expressions
EXAMPLE
Multiply:
SOLUTION
This is the original expression.
Factor the numerators and denominators completely.
Divide numerators and denominators by common factors. Because 3 – y and y -3 are opposites, their quotient is -1.
Blitzer, Intermediate Algebra, 5e – Slide #20 Section 6.1

21. Multiplying Rational Expressions
CONTINUED
Now you may multiply the remaining factors in the numerators and in the denominators. or
Blitzer, Intermediate Algebra, 5e – Slide #21 Section 6.1

22. Multiplying Rational Expressions
Check Point 5
Multiply:
pages
Blitzer, Intermediate Algebra, 5e – Slide #22 Section 6.1

23. Multiplying Rational Expressions
Check Point 6
Multiply:
pages
Blitzer, Intermediate Algebra, 5e – Slide #23 Section 6.1

24. Dividing Rational Expressions
Simplifying Rational Expressions with Opposite Factors in the Numerator and Denominator
The quotient of two polynomials that have opposite signs and are additive inverses is -1.
Dividing Rational Expressions
If P, Q, R, and S are polynomials, where then
Replace with its reciprocal by interchanging its numerator and denominator.
Change division to multiplication.
Blitzer, Intermediate Algebra, 5e – Slide #24 Section 6.1

25. Dividing Rational Expressions
EXAMPLE
Divide:
SOLUTION
This is the original expression.
Invert the divisor and multiply.
Factor.
Blitzer, Intermediate Algebra, 5e – Slide #25 Section 6.1

26. Dividing Rational Expressions
CONTINUED
Divide numerators and denominators by common factors.
Multiply the remaining factors in the numerators and in the denominators.
Blitzer, Intermediate Algebra, 5e – Slide #26 Section 6.1

27. Multiplying Rational Expressions
Check Point 7a
Divide:
Blitzer, Intermediate Algebra, 5e – Slide #27 Section 6.1

28. Multiplying Rational Expressions
Check Point 7b
Divide:
Blitzer, Intermediate Algebra, 5e – Slide #28 Section 6.1

29. DONE