Extra Problems for Unit 5 over sections 4.3 - 5.1

In addition to these problems, make sure you can do and understand: all the assigned problems from the textbook including the review section, quiz 5, activities, and department handouts.

1.) Let	(a.) Find the domain.
(b.) Find the equation of any vertical asymptotes of f .	(c.) Find the coordinates of any ‘holes’ in the graph of f .
(d.) Find the equation of any horizontal asymptotes of f.	(e.) Draw a rough sketch.

2.) Let .

Fill in the blanks:

3.) Suppose f(2) = -3, f(7) = 5, and f is a polynomial function. Suppose g(2) = -3, g(7) = 5, and g is a rational function. Which of the following statements must be true. Circle all that apply.

1. f(c) = 2 for some c Î (-3, 5)
2. f(c) = 2 for some c Î (2, 7)
3. f(c) = 6 for some c Î (-3, 5)
4. f(c) = 6 for some c Î (2, 7)
5. g(c) = 2 for some c Î (-3, 5)
6. g(c) = 2 for some c Î (2, 7)
7. g(c) = 6 for some c Î (-3, 5)
8. g(c) = 6 for some c Î (2, 7)

4.) Use the following graph to fill in the blanks.

y ® _____ as x ® ¥ y ® _____ as x ® -¥ y ® _____ as x ® 1 y ® _____ as x ® 2+ y ® _____ as x ® 2 – y ® _____ as x ® -2+ y ® _____ as x ® -2 –

5.) A right triangle has an area of 40 ft² and a hypotenuse that is 2 feet longer than one of its sides. Let x denote the length of that side. Find the length of its legs. (Recall for a right triangle a² + b² = c² where a & b are the legs and c is the hypotenuse. Also the area is ½ base × height.)

equation used to solve this problem:
solution:

6.) Let f(x) be defined by the following set of ordered pairs: {(-2, 0), (0, 2), (1, -2), (3, 1)} Find the set of ordered pairs that represents f ^-1(x)

(#7 & 8)Solve the following inequalities for x and write the solution in interval notation and graph it.

9.) x is a function of y for which of the following (i.e. the inverse is a function) : _________________

(a.) 3x + 4y = 6		(b.)
(c.)		(d.)
(e.)		(f.) x Î {Jan, Feb, March, April, May, June, July, Aug, Sep, Oct, Nov, Dec} y = A student in this class whose birthday is in month x.
(g)	(h) x = 2(y -2)2		(i) x Î {the students in this class} y = the color of student x’s hair
(j) x3y= 3	(k)		(l)

10.) Find the quotient q(x) and remainder r(x) if  f(x) = 3x⁴ – 2x³ + 2x – 2  is divided by p(x) = x² + 2x - 1.

11.) Completely factor g(x) = 3x³-2x²-19x-6. Hint g(3) = 0.

12.) Find all values of k such that  f(x) = k²x³ – 4kx + 3 is divisible by x-1.

13.) Find the remainder when x¹⁰⁰² + 2x⁸⁴⁷ - 5x⁹⁶ + 3x²² -2 is divided by x + 1 .

14.) Find a polynomial f(x) with real coefficients that has degree 3 with zeros 1 + i and 2 such that f(1) = -3 . Write as a product of linear and quadratic factors that are irreducible over the reals.

15.) Let f(x) be described by the picture at the right: Which of the following is its inverse?

16.)Let . Find the inverse function if it exists. If it doesn’t exist show why.

17.) Let f(x) = x⁴ + 7x³ - 26x² + 28x - 120 .

(a.) State all possible rational zeros of f(x) using the Rational Root Theorem.
(b.) Show that 4 is an upper bound and -12 is a lower bound for the zeros of f(x).
(c.) Use synthetic division to completely factor f(x) over the complex numbers.

18.) Find all the real zeros of f(x) = x⁴ - 4x³ - 7x² + 44x - 44 and state the multiplicity of each.

Answers:

1.) a.) all reals except 2 and -3 b.) x = -3    c.) (2, 4/5)  d.) y = 1
e.)
2.) ∞, -2, 0 respectively
3.) B
4.) ∞ , 2, 0, -∞, -∞, ∞, -∞

7.)  

8.) [-2, 0) ∪ [2, ∞)

13.) (-1)¹⁰⁰² + 2(-1)⁸⁴⁷ - 5(-1)⁹⁶ + 3(-1)²² - 2 = -5 The remainder is -5.

14.) f(x) = 3(x-2)(x² - 2x + 2)

15.) the second one from the left

16.) f ^-1(x) = (3x - 2)^1/3