FUNCTION NOTATION WORD PROBLEMS

Problem 1 :

The value in dollars, v(x), of a certain car after x years is represented by the equation v(x) = 25,000(0.86)^x. To the nearest dollar, how much more is the car worth after 2 years than after 3 years?

1) 2589     3) 15,901

2) 6510     4) 18,490

Solution:

v(x) = 25,000(0.86)^x

After 2 years: x = 2

v(2) = 25,000(0.86)²

v(2) = 18490

After 3 years: x = 3

v(3) = 25,000(0.86)³

v(3) = 15901.4

v(2) - v(3) = 18490 - 15901.4

= 2588.6 more

So, option (1) is correct.

Problem 2 :

The equation to determine the weekly earning of an employee at The Hamburger Shack is given by w(x), where x is the number of hours worked.

Determine the difference in salary, in dollars, for an employee who works 52 hours versus one who works 38 hours. Determine the number of hours an employee must work in order to earn $445. Explain how you arrived at this answer.

Solution:

a) When x = 52 (x > 40)

 w(52) = 15(52 - 40) + 400

= 15(12) + 400

= 580

when x = 38 (0 ≤ x ≤ 40)

 w(38) = 10(38)

= 380

w(52) - w(38) = 580 - 380

= $200

(b) when w(x) = 445, x > 40

445 = 15(x - 40) + 400

45 = 15x - 40(15)

45 = 15x - 600

15x = 645

x = 43

So, an employee must work 43 hours in order to earn $445.

Problem 3 :

Lynn, Jude, and Anne were given the function f(x) = -2x² + 32, and they were asked to find f(3). Lynn's answer was 14, Jude's answer was 4, and Anne's answer was ±4. Who is correct? 

1) Lynn, only     3) Anne, only

              2) Jude, only     4) Both Lynn and Jude

Solution:

f(x) = -2x² + 32

f(3) = -2(3)² + 32

= -18 + 32

f(3) = 14

Therefore, Lynn's answer is correct.

So, option (1) is correct.

Problem 4 :

Amy is purchasing t-shirts for her softball team. A local company has agreed to make the shorts for $9 each plus a graphic arts fee of $85. Write a linear function that describes the cost, C, for the shirts in terms of q, the quantity ordered. Then find the cost of order 20 t-shirts.

Solution:

To find the linear cost function, start from the slope intercept form,

C(x) = mx + b

C(q) = 9q + 85

The cost of order 20 t-shirts

C(20) = 9(20) + 85

= 180 + 85

C(20) = 265

So, the cost of 20 t-shirts is $265.

Problem 5 :

The cost, C, of water is a linear function of g, the number of gallons used. If 1000 gallons cost $4.70 and 9000 gallons cost $14.30, express C as a function of g.

Solution:

The cost of linear function,

C(g) = gx + b

4.7 = 1000x + b ---> (1)

14.30 = 9000x + b ---> (2)

Subtract (1) and (2),

9.6 = 8000x

x = 9.6/8000

x = 3/2500

Put x = 3/2500 in (1),

4.7 = 1000(3/2500) + b

4.7 = 6/5 + b

b = 4.7 - 6/5

b = 3.5

b = 7/2

So, the function is

C(g) = 3/2500g + 7/2

Problem 6 :

If 50 U.S. dollars can be exchanged for 69.5550 Euros and 125 U.S. dollars can be exchanged for 173.8875 Euros, write a linear function that number of Euros, E, in terms of U.S. dollars, D.

Solution:

The linear function expression

E(D) = kD  + b

69.5550 = 50k + b ---> (1)

173.8875 = 125k + b ---> (2)

Subtract (1) and (2),

75k = 104.3325 

k = 104.3325/75

k = 1.3911

Put k = 1.39911 in (1)

69.555 = 50(1.3911) + b

69.555 = 69.555 + b

b = 0

So, the expression is

E(D) = 1.3911 D

Problem 7 :

The Fahrenheit temperature reading (F) is a linear function of the Celsius reading (C). If C = 0  when

F = 32 and the readings are the same at -40°, express F as a function of C.

Solution:

Given, C = 0, F = 32

C = -40, F = -40

b = 32

The linear function expression

y = mx + b

F(C) = 9/5C + 32

Problem 8 :

Recall the example from a previous concept where a student organization sells shirts to raise money. The cost of printing the shirts was expressed as 100 + 7x and for the revenue, we had the expression 15x, where x is the number of shirts.

a. Write two functions, one for the cost and one for revenue.

b. Express that the cost must be less than or equal to $800.

c. Express that the revenue students sell in order to make $1500?

d. How many shirts must the students sell in order to make $1500?

Solution:

a.

The total cost for producing x number of shirts

C(x) = 100 + 7x

The price of each shirts is $15.

Revenue = price per shirt  × number of shirts

 R(x) = 15x

b.

100 + 7x ≤ 800

c.

1500 = 15x

d.

15x = 1500

x = 100

So, the students should sell 100 shirts in order to make $1500.

Problem 9 :

If the value of a car t years after purchase is given by

V(t) = 28000 - 4000t dollars.

a)  find V(4) and state what V(4) means

b)  find t when V(t) = 8000 and explain what this represents

c)  find the original purchase price of the car.

Solution :

a) V(4)

V(t) = 28000 - 4000t

When t = 4

V(4) = 28000 - 4000(4)

V(4) = 28000 - 16000

V(4) = 12000

After 4 years, the cost of the car is 12000.

b)  V(t) = 8000

8000 = 28000 - 4000t

20000 = 4000t

t = 5

V(t) = 8000 represents after how many years, the cost of the car will be 8000.

c) V(t) = 28000 - 4000t

From the given equation, we understand that 4000 is the depreciation rate.

when t = 0

V(0) = 28000 - 4000(0)

= 28000

So, the initial price is 28000.