Practice Exam 2

Exam 2 covers sections R6, 2.4, 2.7, 3.1, 3.2, 3.3, 3.4, 4.1, 4.2, 4.3. The actual Exam 2 will be shorter than the practice test. The format of Exam 2 will be similar to that of the practice test.

Part I -- No calculators allowed.

Some of the problems below seem like multiple-choice problems, but indeed they are not. You will have to show your work, no credit will be given without work. Partial credit is possible, similarly as on Exam I. In all problems, your explanations should consist of complete sentences and correct mathematical formulas.

1. Simplify

(A) (B) (C) Can't be simplified (D)

2. Simplify.

(A) (B) (C) (D)

3. Simplify

(A) (B) (C) (D)

4. Write as one quotient and simplify

(A) (B) (C) (D) Can't be done

5. Assume c >0, d> 0. Simplify

(A) (B) (C) (D)

6. Assume a>0, b>0. Simplify

(A) (B) (C) (D) None of the above

7. Find the following logarithms, if defined. If not, write "undefined".

(a) =

(b) =

(c) =

(d) =

(e) =

(f) =

8. Simplify

(a) =

(b) =

(c) =

9. Without using your grapher, find all vertical and horizontal asymptotes of .

10. Does the function have any vertical asymptotes? Explain!

11. Solve the following inequalities. Represent your solutions graphically on the number line.

(a)

(b)

Part II -- You can use your calculators if you wish. In all problems, your explanations should consist of complete sentences and correct mathematical formulas.

12. Find the center, the radius, and the equation in the standard form for the circle

.

HINT: Complete the square in x terms and in y terms.

13. By completing the square, find the vertex of the parabola .

14. Solve the following equation by completing the square, without the quadratic formula:

.

15. Solve the following inequalities:

(a) (b)

16. Factor completely into polynomials of degree one

17. Is the following function one to one? Explain!

18. Is the following function one to one? Explain!

19. Find a rational function that has y=2 as its horizontal asymptote and x=3 as a vertical asymptote.

20. For the function

find all horizontal and vertical asymptotes. Graph the function.

21. Using a grapher, estimate all zeros of the polynomial

.

22. Let , . Divide P(x) by d(x) using long division . Write P(x) in terms of the divisor, the quotient, and the remainder.

23. For the following functions f(x), find the inverse (x) if exists. If the inverse exists, graph both functions in one coordinate system.

(a) , domain: all x. (b) , domain: all x (c) , domain:

24. Find the inverse function. You don't have to graph anything.

(a) (b)

25. Let , . Find the composite functions f(g(x)) and g(f(x)). You don't have to simplify anything.

26. For each function h(x) below, find at least two ways of representing h(x) as a composition h(x)=f(g(x)):

(a) (b)

27. Graph and in one coordinate system. How are the two graphs related?