**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**