College Algebra Problem Set

MATH 17 CDUV2 PROBLEM SET 1 (to be passed on August 31, 2007) Directions: In a short bond paper/s, answer the following problems completely, orderly and neatly. Show all necessary solutions and double check your answers to avoid careless mistakes. Also, avoid erasures (erasures in any kind will make that particular item void). Work as a group and make sure that every member in a group understands the solution and answer. I. Composite Functions: The functions f and g are defined. Determine the following (a) f  g ; (b) g  f ; (c) f  f ; (d) g  g and give their respective domains. 1. f ( x ) = x ; g ( x ) = −

1 x

2. f ( x ) = x 2 − 1; g ( x ) = x − 1 3. f ( x ) =

x+3 ; g ( x) = x + 3 x−2

II. Rational Functions: Determine the domain, range, vertical and horizontal asymptotes of the following functions and draw a sketch of its corresponding graphs. 1. y = f ( x ) = −

1 x

x 2 − 4x + 3 6. y = g ( x ) = x2 −1

2. y = g ( x ) = −

1 x2

7. y = h( x ) =

x 2 − x − 12 x2 + x − 6

3. y = h( x ) =

2x + 4 4 − 2x

8. y = j ( x ) =

x 2 − 25 x+5

4. y = j ( x ) =

4x x −4

9. y = f ( x ) =

x−5 x − 8 x + 12

5. y = f ( x ) =

x +1 2 x + 4x

10. y = g ( x ) =

2

2

3x 2 x2 + 4

III. Absolute Value Functions: Find the domain, range of the following functions and sketch its corresponding graphs. 1. y = f ( x ) = x − 6 − 3

2 4. y = f ( x ) = x + 4 x − 12

2. y = g ( x ) = −2 5 − x + 1

5. y = g ( x ) =

x −5

6. y = h( x ) =

1− x +1 x +1

3. y = h( x ) =

4 + 2x 3

−2

IV. Conditional Functions: Find the domain, range of the following functions and sketch its corresponding graphs.  x 2 − 1 if x > −1 1. y = f ( x ) =   x + 1 if x ≤ −1

 25 − x 2 if − 5 < x < 3 4. y = f ( x ) =  if − 3 < x ≤ 5 [ x ]

 4 − x if x < 4 2. y = g ( x ) =  − x 2 − 16 if x > 4

 x −1  5. y = g ( x ) =  2  x −3

− x − 5 if x < −5  2 3. y = h( x ) =  25 − x if − 5 ≤ x < 4 2 x − 3 if x > 4 

− 1 if x < 0  6. y = h( x ) =  0 if x = 0  1 if x > 0 

if x ≤ 3 if x > 3

V. Inverse Functions: 1. Given the real constants a, b, c, d. Solve for the inverse of F ( x ) =

ax + b . cx + d

2. Cite an example of a function f, other than the identity function, such that f ( x ) = f

−1

( x) .

VI. Functions as Mathematical Models: Define all the variables that is used and solve the following problems. 1. A travel agency offers an organization an all-inclusive tour for $800 per person if not more than 100 people take the tour. However the cost per person will be reduced $5 for each person in excess of 100. How many people should take the tour in order for the travel agency to receive the largest gross revenue, and what is the largest gross revenue? 2. The graph of an equation relating the temperature reading in Celsius degrees and the temperature reading in Fahrenheit degrees is a straight line. Water freezes at 0 degrees Celsius and 32 degrees Fahrenheit and water boils at 100 degrees Celsius and 212 degrees Fahrenheit. (a) Express the number of degrees in the Celsius temperature as a function of Fahrenheit temperature; (b) and solve for the inverse of this function. 3. A particular lake can support a maximum of 14,000 fish and the rate of growth of the fish population is jointly proportional to the number of fish present and the difference between 14,000 and the number present. (a) Let f ( x ) fish per day be the rate of growth when there are x fish present, write an equation defining f ( x ) . (b) What is the domain of the function f ? 4. A carpenter can sell all the end tables that are made at a price of $64 per table. If x tables are built and sold each week, then the number of dollars in the total cost of the week’s production is x 2 + 15 x + 225 . (a) Express the number of dollars in the carpenter’s weekly profit as a function of x. (b) How many tables should be constructed each week in order for the carpenter to have the greatest weekly total profit? (c) What is the greatest weekly total profit? -END OF PROBLEM SET 1-