Answers to selected Homework Exercises:
Section 1.1,
#8 Yes, it’s a function. The domain is
[−3, 2] and the range is [−2, −2] ∪ [(0, 3]
#19 see answers in the back of the text.
#22
2+h
f (2 + h) =
3+h
x+h
f (x + h) =
x+h+1
x+h
x
−
f (x + h) − f (x)
= x+h+1 x+1
h
h
1
=
(x + h + 1) (x + 1)
#24
Set the denominator equal to
1
zero to get the x’s not in the domain
x2 + 3x + 2 = 0
(x + 1)(x + 2) = 0
x = −1, −2
Therefore the domain is (-∞, −2) ∪
(−2, −1) ∪ (−1, ∞)
#26
The domain is [0,4]
#36 Let’s do live
#58
even
#60
odd
Appendix A
−3
or
(∞,
#12
x≤ −3
2
2 ]
#14
x > 12 , or ( 12 , ∞)
#16
(−1, 4]
#22
[−3, −1] ∪ [2, ∞)
2
Lines
The first point we want to make about
lines is that they have a slope. We commonly
let m be the slope symbol. A positive slope,
m > 0,
increases as we go left to right. A negative
slope, m < 0,
decreases as we go left to right. A slope
of zero, m = 0, is a horizontal line and a
vertical line has an infinite slope.
The equation for the slope between two
points, (x1, y1) and (x2, y2),of a line is
rise y2 − y1
=
m=
run x2 − x1
3
The slope between two points is called the
secant slope.
A steeper slope means a larger magnitude
m.
We will use two types of equations for a
line.
• Point-Slope Equation of a line, y − y1 =
m(x − x1)
• Slope-Intercept Equation of a line, y =
mx + b
Both forms have their advantages. When
you are trying to make the equation of a line
from some given information, the Point-Slope
version is the best. The Slope-Intercept
form is good if you want to graph the line.
Remember that you can change one form into
the other.
Example
Q?
Find the equation of the line
through the point (2,3) with a slope of 5.
4
A.
Use the Point-Slope equations
with m = 5 and (x1, y1) = (2, 3).
y − 3 = 5(x − 2)
We can change that Point-Slope equation
into a Slope-Intercept equation by multiplying
it out.
y − 3 = 5(x − 2)
y − 3 = 5x − 10
y = 5x − 7
This is the same line but written in the SlopeIntercept form. Let’s show why the second
way, the Slope-Intercept way, to write a line
is good. With y = mx + b the slope is still
m and the y-intercept is b. Therefore if we see
y = 5x − 7 we note that the slope is m = 5
and the y-intercept is b = −7. This lets us
sketch the line.
5
10
5
-1
0
1
2x
3
4
-5
-10
Some lines don’t fit nicely into these two
forms.
• A vertical line has an infinite slope and an
equation of the form x = a, where a is the
x-intercept.
• A horizontal line has a slope of zero and
has the equation y = b where b is the
y-intercept.
Parallel and Perpendicular Slopes
If two lines are parallel then they have
the same slope. Parallel lines don’t cross,
they don’t intersect unless they are the same
line. When the lines are written in the SlopeIntercept form it is easy to see if they are
parallel.
6
Perpendicular lines intercept at a right
angle. Another word for perpendicular
is orthogonal or normal. The slopes of
orthogonal lines are negative reciprocals to
each other such as m and −1
m.
Example
Q?
Given the lines y = 5x − 4, y =
3x − 2, y = 5x + 9, y = −1
3 x + 7, 2y − 10x =
−8 which are parallel? Which ones are
perpendicular?
Example
Q?
Find the equation of the line that
is perpendicular to the line y = 12 x − 3 and
that goes through the point (1, 5). At what
point do the two lines intersect?
The method of equating the two equations
to find their points of intersection works for
all types of functions, not just lines.
Appendix B #1, 3, 5, 11 thru 24.
Not to be handed in.
7