MATH 107 FINAL EXAMINATION

MATH 107 FINAL EXAMINATION

This is an open-book exam. You may refer to your text and other course materials as you work on the exam, and you may use a calculator. You must complete the exam individually. Neither collaboration nor consultation with others is allowed.
Record your answers and work on the separate answer sheet provided.

There are 30 problems.

Problems #1–12 are Multiple Choice.
Problems #13–21 are Short Answer. (Work not required to be shown)
Problems #22–30 are Short Answer with work required to be shown.

MULTIPLE CHOICE

1. Determine the domain and range of the piecewise function.

A. Domain [–1, 3]; Range [–1, 1]
B. Domain [–1, 3]; Range [–2, 2]
C. Domain [–2, 1/2]; Range [1/2, 3/2]
D. Domain [–2, 2]; Range [–1, 3]

2. Solve: 14 −5x = − x 2. ______
A. –7
B. 7/2
C. –7, 2
D. No solution

3. Determine the interval(s) on which the function is decreasing.

A. (–¥, –1)
B. (– 4.5, – 1) and (2.5, ¥)
C. (– 2, 2)
D. (–¥, – 3) and (1, ¥)

4. Determine whether the graph of y = 3×2 − 5 is symmetric with respect to the origin, the x-axis, or the y-axis.

A. symmetric with respect to the x-axis only
B. symmetric with respect to the y-axis only
C. symmetric with respect to the origin only
D. not symmetric with respect to the x-axis, not symmetric with respect to the y-axis, and
not symmetric with respect to the origin

5. Solve, and express the answer in interval notation: | 9 – 6x | ³ 3.

A. [1, 2]
B. [1, ¥)
C. (–¥, 2] È [1, ¥)
D. (–¥, 1] È [2, ¥)

6. Which of the following represents the graph of 9x + 4y = 36 ?

7. Write a slope-intercept equation for a line parallel to the line x – 5y = 9 which passes through
the point (–10, 3).

A. Y = -1/5X+3
B. Y = -1/5X+3
C. Y = -1/5X+5
D. y = 5x + 53

8. Does the graph below represent a function and is it one-to-one?

A. It is a function and it is one-to-one.
B. It is a function but not one-to-one.
C. It is not a function but it is one-to-one.
D. It is not a function and it is not one-to-one.

9. Express as an equivalent expression: 3 log y + log 1 – log (x – 8)

A. log(3y)
log(x-8)

B. log(y3/x-8)

C. log(3y+1/x-8)

D. log(3y+9-x)

10. Which of the functions corresponds to the graph?

A. f(x)= e-x+4
B. f(x)= e-x+3
C. f(x)= ex+3
D. f(x)= ex+4

11. Suppose that for a function f , f has exactly one x-intercept.

Which of the following statements MUST be true?

A. f is an invertible function.
B. f is a linear function.
C. The equation f (x) = 0 has exactly one real-number solution.
D. There is exactly one point on the graph of f which has an x-coordinate of 0.

12. The graph of y = f (x) is shown at the left and the graph of y = g(x) is shown at the right. (No
formulas are given.) What is the relationship between g(x) and f (x)?

A. g(x) = f (x – 1) + 2
B. g(x) = f (x + 1) – 2
C. g(x) = f (x – 2) + 1
D. g(x) = f (x + 2) + 1

SHORT ANSWER:

13. Multiply and simplify: (6 + i)(7 + 3i).

Write the answer in the form a + bi, where a and b are real numbers.

14. Solve, and write the answer in interval notation: x+4/x-1>0

15. A bowl of soup at 170° F. is placed in a room of constant temperature of 60° F. The temperature T of the soup t minutes after it is placed in the room is given by

T(t) = 60 + 110e-0.075t

Find the temperature of the soup 6 minutes after it is placed in the room. (Round to the nearest degree.)

16. Find the value of the logarithm: log5(1/25)

17. Solve: 89x−5 = 64

18. Suppose $7,200 is invested in an account at an annual interest rate of 6.8% compounded continuously. How long (to the nearest tenth of a year) will it take the investment to double in size?

19. Let f (x) = x2 + 6x + 8.

(a) Find the vertex.
(b) State the range of the function.
(c) On what interval is the function decreasing?

20. Consider the polynomial P(x), shown in both standard form and factored form.

P(x)= 1/3x<sup>4</sup>+1/3x<sup>3</sup>-7/3x<sup>2</sup>-1/3x+2=1/3(x+1)(x-1)(x-2)

(a) Which sketch illustrates the end behavior of the polynomial function?

(b) State the zeros of the function.

(c) State the y-intercept.

(d) State which graph below is the graph of P(x).

21. Let f(x)=3x<sup>2</sup>+3/x<sup>2</sup>-4

(a) State the domain.
(b) State the vertical asymptote(s).
(c) State the horizontal asymptote
(d) Which of the following represents the graph of f(x)=3x<sup>2</sup>+3/x<sup>2</sup>-4?

SHORT ANSWER, with work required to be shown, as indicated.

22. Let f (x) = x + 3 and g(x)= 7 − x .

(a) Find (f/g)(-9). Show work.

(b) Find the domain of the quotient function f/g. Explain.

23. Points (–1, 8) and (9, 10) are endpoints of the diameter of a circle.

(a) What is the length of the diameter? Give the exact answer, simplified as much as possible.
Show work.
(b) What is the center point C of the circle?
(c) Given the point C you found in part (b), state the point symmetric to C about the y-axis.

24. Find the equation for a line which passes through the points (3, 7) and (6, 1). Write the equation in slope-intercept form. Show work.

25. Ron, a resident of Metropolis, pays Metropolis an annual tax of $60 plus 1.8% of his annual income. If Ron paid $1,032 in tax, what was Ron’s income? Show work.

26. Let f (x) =2x<sup>2</sup>+9 and g(x)= x-6

(a) Find the composite function ( f o g)(x) and simplify. Show work.
(b) Find ( f o g ) (−1) . Show work.

27. Find the exact solutions and simplify as much as possible: 10x<sup>2</sup>+9 =20x. Show work

28. Given the function f(x) = 1/4x -8, find a formula for the inverse function. Show work

29. Donut Delights, Inc. has determined that when x donuts are made daily, the profit P, in dollars, is given by

P(x) = –0.001 x<sup>2</sup>+2.9x – 1000

(a) What is the company’s profit if 900 donuts are made daily?
(b) How many donuts should be made daily in order to maximize the company’s profit? Show work.

30. Solve: x+9 / x+6 + 36/ x<sup>2</sup>-36 =0 Show work

Here’s the SOLUTION

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