Math 135  Spring 2012
Key Number Method WorksheetSolutions
Solve using the key number method and write the solution in interval notation:
1. x2 + 4x + 4 < 0
Answer Factoring we have
x2 + 4x + 4 < 0 (x + 2)(x + 2) < 0
There is no denominator and hence no
Math 135  Spring 2012
Functions: The Inverse  Solutions
1. In the Functions: Examples worksheet from Week 5 do the following:
(a) Determine whether each function is onetoone.
Answer The onetoone functions, i.e. the ones which pass the horizontal lin
 Spring 2011
1
lr a
3',The square.filncticin: f (x) :'12.
Domain ll (oo, m
Range ll
Intervals of Increase
Intervals of Decrease
Ttrrning Points
Local Maxima
Local Minima
Global Maxima
Global Minima
[0,
[0,
*)
*)
(*,
ll
ll r : 0
ll
a
O]
I
I
I
\
I
I
,l
t
 Spring 2012
1
(b) g(r)
:
3r2

6r
*2
Answer Completing the square we obtain
3r2  6r
*2 : 3(r' +2r) +2
: 3(r'*2r*1) +2+3
: 3(r+t)2+s
and we read off the desired information:
. a: 3 < 0 so the parabola opens down.
. (1,5) is the vertex.
. r : l i
Math 135  Spring 2012
Absolute Value Worksheet  Solutions
1. Draw the interval (2, 3] on the number line.
Answer: See the last page.
2. Arrange from least to greatest: 2, ,  2,  1, 1. Use the symbols < and
.
Answer:
2 <  1 < 1 <  2 < 
3. Simp
Math 135  Spring 2012
Functions: Arithmetic  Solutions
Compute and simplify:
1. f (x) = x2 + x + 1, f (x+h)f (x) =
h
Answer
(x + h)2 + (x + h) + 1 (x2 + x + 1)
f (x + h) f (x)
=
h
h
2
2
x + 2xh + h + x + h + 1 x2 x 1
=
h
2xh + h2 + h
=
h
= 2x + h + 1
2.
Math 135 Spring 2012
Linear Functions Worksheet
Find the linear functions satisfying the given conditions:
1. f (3) = 2 and f (3) = 4.
We have the points (3, 2) and (3, 4). The slope of the line connecting these two
points is:
a=
y2 y1
2 (4)
6
=
= =1
x2
Math 135  Spring 2012
Integer Exponents  Solutions
Recall the rules for working with exponents. Let x, a, b be any real numbers. Then:
x0 = 1, x = 0
xa xb = xa+b
xa
xb
= xab
(xa )b = xab
x
y
a
xa
ya
=
x1 =
1
x
(xy)b = xb y b
Simplify:
1.
3zx3 y 4
6