Download Sample Exam Key

Name:
Mathematical Investigations
An Opportunity to Demonstrate Knowledge – Without your calculator
1.
Generalize each pattern:
a. {(2, 7), (1, 4), (0,1), (1, 2), (2, 5), (3, 8)}
(x, y)
b.

y  3x  1
 (3,5),1 ,  (3,5),1 , (2,5),1 , (2,10), 2 , (3,15),3 ,  (20,30),10 ,
 ( x, y), z  z  greatest common factor of x and y
Fcn Exam p. 1
Rev. F11
For problems 3, if the representation is a function, give its domain and range. If it is not a
function, do not give its domain and range; instead, indicate why it is not a function. Be specific!
3.
(3, 2), (7, 4), (9,5), (7,11), (0,13)
Function?
Yes
No
X
The element 7 in the domain correspond to two different elements in the range(4 and 11).
By definition of a function, each element in the domain must correspond to a unique
element in the range.
4.
5.
If f = { (–4,6), (–1,3), (0,5), (2,7), (7,–4) } and g = { (–4,2), (0,–4) ,(7,–1), (5,7) (6, 9)}
a.
Determine the function: f g = {(4, 7), (0, 6), (7,3), (5, 4)}
b.
Determine f g (7) = 3
c.
If f g ( x)  4 , then g ( x ) 
d.
Determine the function f
7___ and x =
5
.
f = {(2, 4), (7, 6)}
x4
. Give your answer using interval notation and graph
2x  6
your answer on the number line.
Find the domain of h( x) 
x4
 0.
2x  6
The critical points are: x = 4, 3 .
We label these points on the number line using an open circle for x = 3 since this is not in the
domain, and a closed circle for x = 4.
The critical points divide (partition is a nice word) the number line into three sections. Numbers
less than x = 3 work in the inequality, as do numbers greater than (or equal to) x = 4 .
We need to solve
Domain = (, 3) [4, ) in interval notation.
Fcn Exam p. 2
Rev. F11
6.
If f ( x)  x 2  2 x  48 ,
a.
find all a such that f a   0 .
f (a)  a 2  2a  48  (a  6)(a  8), so
f (a)  0  a  6 or 8
b.
find all b such that
f  b  6 = 0.
In order for f  b  6 = 0, we must have (from part a), b  6  6 or b  6  8.
Therefore, b  0 or b  14.
7.
Given the function f (t )  (t  2,3t  5) ,
a.
Find an equation for y in terms of x. (that is, eliminate the parameter t).
x  t  2  t  x  2. therefore, y  3t  5  3( x  2)  5  y  3x  1
b.
Write the domain and range of f.
Domain = {t t is any real number}
8.
Range = {( x, y) x and y are real numbers}
Suppose that Rachel is renting calculator batteries where the fixed cost of the rental is $2
plus $0.25 per minute for each minute of use. Since she needs them back by the end of
class, batteries can only be rented out for 15 minutes. Note: fractions of a minute generate
a charge for an additional full minute.
a. Write an equation that gives the cost of rental, y, in terms of the number of minutes the
batteries are rented.
y  .25t  2 , t represents the rental time in minutes.
b. Find the domain and range of the function you found in part a.
Domain = {t 0  t  15}
Range = {2.25, 2.50,....5.75}
Fcn Exam p. 3
Rev. F11
9.
Given f ( x)  x 2  5 and g ( x)  x  4 .
a.
Determine f g ( x) =
b.
Give the domain of f g ( x) . Domain = {x x  4}
c.
Use the axes below to sketch the graph of f g ( x) .

x4

2
 5  x  1 , provided x  4
Fcn Exam p. 4
Rev. F11