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Math 121 Exam #1 Review Fall 2012

1. Determine the domain of the following functions. Write your answers using interval notation:

a) ( ) 8 3f x x= +

b) ( ) 8 3f x x= −

c) 6 7

3 2

x y

x

+ =

−

d) 4 11y x= −

e) 2( ) 3g t t= −

2. Find the domain and range of the following functions.

3. Given 2

4 ( )

1 f t

t =

+ . a) Find ( 2)f − b) Find the values of t such that ( ) 2f t = .

c) Find the Avg. Rate of Change in f from� = 0 to � = 3.

4. Compute the Avg. Rates of Change in the World Population using the data from the table below.

Year Population in

billions

Avg. Rate of Change

1850 1.26

1900 1.65

1950 2.52

2000 6.09

a) What are the units for the Avg. Rates of Change in this problem? b) Is the World Population a linear function of the year? Explain.

4 5 ( )

2 5

x g x

x

− =

+( ) 2

( ) 2 3f x x= − − +

5. Find a formula for each of the following function from the verbal description: a) Starting with an input of x, add 3 then multiply by 5 and then subtract 2. b) Starting with an input of x, multiply by 2, then add 3 then take the square root. 6. Write a verbal description for each of the following functions given by formula:

a) ���� = 3 4�� − 1� − 1

b) ���� = ��� �����

7. For each of the following find an equation in slope-intercept form.

a) The line with slope � and passes through �3,−4�

b) The line that passes through ( 1, 2)− and ( 6, 5)− − .

c) The vertical line that passes through ( 4, 7)− .

d) The horizontal line that passes through ( 8, 5)− .

e) The line parallel to 1 3

5y x= − and passing through ( 1, 7)− .

f) The line perpendicular to 1 3

5y x= − and passing through ( 1, 7)− .

8. The speed of sound in air changes linearly with the temperature of the ambient air. When the

temperature, T , is 32 F o

the speed of sound, S , is ft sec

1087.5 . On a nice spring day, when the

ambient temperature is 70 F o

, the speed of sound is ft sec

1129.3 .

a) Find the linear function which gives the speed of sound, S , as a function of the ambient temperature, T (in Fahrenheit).

b) What is the practical meaning of the slope of this function? Be sure and include units in your

explanation.

c) Find the speed of sound at an ambient temperature of 61 Fo . Use correct units in your answer. 9. A union has negotiated a plan to provide uniform pay raises for its members. A person whose starting

salary is $36000 will be making $42,000 four year later. Find a linear function to model the salary

���� of this person after t years. How much will this person be earning after 7 years? How many years will it take for this person salary be $72,000?

10. A water tank which starts with 1400 gal of water is being drained at the rate of 8 gal/min. Find a

linear function which model the amount ���� of water in the tank after t minutes. How long will it take to empty the tank?

(a, 0)

(u, v)

x

y

-6 -4 -2 2 4 6 -1

1

2

3

4

5

x

y

(-6, 2) (2,2)

(6, 1)

(2, 4)

11. Find the equation for the line shown below. Your equations will involve the constants given on the

graph. Be sure to give your answer in slope -intercept form. 12. Solve the following systems of linear equations.

a) 5 3 3.3

2 0.7

x y

x y

+ =

+ = b)

1

3 5 5

2 3 5

3 4

x y

x y

− + =

− = −

c)

2 4 7

3 2

x y

x y

− =

= +

13. A small company manufactures high-end mountain bicycles that cost of $750 each to produce. The company has a fixed monthly cost of $5400 and sells their bicycles for $1200 each.

a) Find the linear function C that gives the total cost of producing x bicycles a month. b) Find the linear function R that gives the total revenue for selling x bicycles a month. c) Will the company make a profit if they make and sell 10 bicycles in a month? d) Find the break-even point, that is, find the number x of bicycles they need to produce and sell in a

month so that their cost equals their revenue.

14. Construct a piecewise linear function for the graph of ( )f x below.

15. The tax code in a particular state requires residents to pay 7% income tax on the first $40,000 they

earn, and 9% on the portion of income above $40,000 level. Find a piecewise linear function that gives the amount of tax owed as a function of income. What is the tax on $40,000? What is the tax on $40,001? Does your answer make sense? If a person pays $3700 in tax what is the income?

16. The owner of a new restaurant is making plans to buy tables. She plans to have tables that seat 4

people each, tables that seat 6 people each and tables that seat 8 people. The local fire code indicates that the restaurant can set at most 108 people. A professional restaurant association suggests that the number of tables which seat 4 people should twice as many as the number that seats 6. And the number of table which seat 8 people should be half as many of the number which seat 6 people.

Setup a system of 3 linear equations which can be used to determine the number of each type of table the owner should buy. Be sure to clearly indicate and define your variables. Solve this system.

17. (Challenge problem from Economics) The diagram to the

right shows a simple model of an “economy” which consist of two industries and consumers. Let x be the number of units output by Industry A and y the number of units output by industry B. From the diagram, we see that consumers demand 1000 units from A and 780 from B. However there is also a demand on A from B. Each unit y which B output requires .1 of a unit from A. Similarly each unit x which A outputs requires .2 of a unit from B. The total demand on A is given

by 1000 .1x y= + , the output from A is 1000 plus .1 of the

output from B. This equation can be rewritten as

.1 1000x y− = . Find the demand equation on B and then

solve this system to determine what the outputs from Industry A and Industry B should be.

18. (Challenge problem from Engineering) In the study of heat transfer a useful notion for estimating an unknown temperature is to take the average of the temperatures of the four closest points. Consider the figure at the right. Suppose you want to find the temperatures x,y,z. Applying the notion, we could setup a system of three equation and solve it to estimate x,y,z. For example the equation to estimate the

temperature at x is: 200 200

4

y z x

+ + + = , this is just the equation you

get when you write x equal to the average of the 4 closest points. This

equation can be rewritten as 4 400x y z− − = . Setup similar equations

for estimating y and z. Then solve this system of 3 equations.