# Algebra

College Algebra MATH 107 Fall, 2015, V3.1

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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. 1. ______

A. Domain [–1/2, 0]; Range [–1, 0] B. Domain [–3, 1]; Range [–1, 3] C. Domain [–1, 3]; Range [–3, 1] D. Domain [–1, 1]; Range [–1, 3]

2. Solve: 45 4x x− = − 2. ______

A. –9, 5 B. –9 C. 15 D. No solution

2 4 -4

-2

-4

2

4

-2

College Algebra MATH 107 Fall, 2015, V3.1

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

A. ( )5.0, −∞− and (5, 6.5) B. (– 0.5, 3) C. (–2, 2) D. (1, 5)

4. Determine whether the graph of 2

3xy += is symmetric with respect to the origin,

the x-axis, or the y-axis. 4. ______

A. symmetric with respect to the origin only B. symmetric with respect to the x-axis only C. symmetric with respect to the y-axis 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: | 8 – 6x | ≥ 2. 5. ______

A. [1, ∞) B. [1, 5/3]

C. (–∞, 5/3] ∪ [1, ∞)

D. (–∞, 1] ∪ [5/3, ∞)

College Algebra MATH 107 Fall, 2015, V3.1

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6. Which of the following represents the graph of 3x − 8y = 24 ? 6. ______ A. B.

C. D.

College Algebra MATH 107 Fall, 2015, V3.1

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7. Write a slope-intercept equation for a line parallel to the line x – 2y = 6 which passes through the point (10, – 4). 7. ______

A. 1

1 2

y x= − +

B. 1

9 2

y x= −

C. 1

4 2

y x= −

D. 2 4y x= − −

8. Which of the following best describes the graph? 8. ______

A. It is the graph of a function but not one-to-one. B. It is the graph of a function and it is one-to-one. C. It is the graph of an absolute value relation.

D. It is not the graph of a function.

College Algebra MATH 107 Fall, 2015, V3.1

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9. Express as a single logarithm: 6 log x + log 1 – log y 9. ______

A.

6

1 log

x

y

 +    

B. ( )log 6 1x y+ −

C. ( )6log 1x y+ −

D. 6

log x

y

     

10. Which of the functions corresponds to the graph? 10. ______

A. ( ) 1xf x e−= −

B. ( ) 1xf x e−= +

C. ( ) xf x e= −

D. ( ) 2xf x e= −

College Algebra MATH 107 Fall, 2015, V3.1

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11. Suppose that for a function f , the equation f (x) = 0 has no real-number solutions. Which of the following statements MUST be true? 11. ______ A. f (x) > 0 for all x in the domain of f. B. f is an invertible function. C. f has no x-intercepts. D. f has no y-intercepts. 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)? 12. ______ y = f (x) y = g(x) A. g(x) = f (x – 1) + 2 B. g(x) = f (x – 2) + 1 C. g(x) = f (x + 1) + 2 D. g(x) = f (x + 2) + 1

2 4 -2 -4

-2

-4

2

4

2 4 -2 -4

-2

-4

2

4

College Algebra MATH 107 Fall, 2015, V3.1

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SHORT ANSWER: 13. Multiply and simplify: (6 + 7i)(2 + 3i). Write the answer in the form a + bi, where a and b are real numbers. Answer: ________

14. Solve, and write the answer in interval notation: 7

0 2

x

x

+ ≤

15. Water initially at 200° F. is left in a room of temperature 60° F to cool.

After t minutes, the temperature T of the water is given by T(t) = 60 + 140e – 0.096 t

Find the temperature of the water 20 minutes after it is left to cool. (Round to the nearest tenth of a degree.) Answer: ________

16. Find the value of the logarithm: 7 1

log 49

     

17. Solve: 7 15 25x − = . Answer: ________

18. Suppose \$4,000 is invested in an account at an annual interest rate of 3.8% compounded continuously. How long (to the nearest tenth of a year) will it take the investment to double in size? Answer: ________

19. Let f (x) = x 2 −−−− 6x + 2.

(a) Find the vertex. Answer: ________ (b) State the range of the function. Answer: ________ (c) On what interval is the function increasing? Answer: ________

College Algebra MATH 107 Fall, 2015, V3.1

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

4 3 21 3 3 13 1 ( ) 6 ( 2)( 1)( 3)( 4)

4 2 4 2 4 P x x x x x x x x x= − + + − = + − − −

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

(b) State the zeros of the function. Answer: ________________

(c) State the y-intercept. Answer: ________________

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

GRAPH A GRAPH B

GRAPH C GRAPH D

A.

B. C. D.

vvvv

vvvv vvvv vvvv

College Algebra MATH 107 Fall, 2015, V3.1

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21. Let 2

5 ( )

9

x f x

x =

− .

(a) State the domain. Answer: _________________

(b) State the horizontal asymptote. Answer: _________________

(c) State the vertical asymptote(s). Answer: _________________

(d) Which of the following represents the graph of 2

5 ( )

9

x f x

x =

GRAPH A. GRAPH B.

GRAPH C. GRAPH D.

College Algebra MATH 107 Fall, 2015, V3.1

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

22. Let f (x) = x − 8 and 5)( −= xxg .

(a) Find )9( 

  

g

f . Show work.

(b) Find the domain of the quotient function g

f . Explain.

23. Points (8, –1) and (6, –5) 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 (–7, 2) and (–9, 4). Write the equation in slope-intercept form. Show work.

25. Charles, a resident of Metropolis, pays Metropolis an annual tax of \$80 plus 1.2% of his annual income. If Charles paid \$776 in tax, what was Charles’ income? Show work.

26. Let f (x) = 3x 2 + 9 and g(x) = x – 4.

(a) Find the composite function ))(( xgf o and simplify. Show work.

(b) Find ( ) (3)f go . Show work.

27. Find the exact solutions and simplify as much as possible: 7x 2 − 2 = 2x. Show work.

28. Given the function 1

( ) 2 7

f x x= + , find a formula for the inverse function. Show work.

College Algebra MATH 107 Fall, 2015, V3.1

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29. The fuel efficiency F for a particular car is given by

F(x) = –0.018 x 2 + 1.548 x + 5.8

where x is the speed of the car in miles per hour (mph) and F(x) is the corresponding fuel efficiency in miles per gallon (mpg). (a) What is the fuel efficiency if the car’s speed is 50 mph? (b) What speed will yield the maximum fuel efficiency? Show work.

30. Solve: 2

1 24

3 9

x

x x

− =

+ − . Show work.

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