Problem Set 1
Math 1090-001 (Spring 2001)
Due: Friday Jan. 26, 2001
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1. Determine the domain, codomain and range of each of the following functions:
a)
b)
c)
and f (x) = −x.
f : [0, ∞) −→
f:
2
and f (x, y) = −(xy)2 . Here,
−→
2
denotes the set of all ordered pairs (x, y) of real numbers.
and
f : [0, 1] ∪ [2, 3] −→
f (x) =
2. a)
b)
2x, x ∈ [0, 1]
−3x, x ∈ [2, 3]
√
Suppose the action of f is given by f (x) = 2x + 1, where x is a real number. What is the largest possible
domain of f ?
Hint: For what set of values of x ∈
is f (x) undefined? The largest possible domain of f is simply the
complement of that set in .
Suppose the action of f is given by f (x) =
domain of f ?
3. Let S :
−→
where x is a real number. What is the largest possible
and S(n) = 12 n(n − 1).
a)
What are the domain and codomain of S?
b)
Evaluate
i)
ii)
iii)
iv)
2x+3
3x+4 ,
S(1)
S(2)
S(n − 1)
S(n) − S(n − 1)
4. Do Problems 1.2.23 to 1.2.30 in the textbook. The domain and range of each given function are not stated in
these problems. Take it to be the largest possible one such that the resulting function can be defined. State this
largest possible domain.
5. Find the coordinates of the midpoint of the line segment joining the points (x1 , y1 ) and (x2 , y2 ).
6. Explain why each of the following curves does or does not represent y as a function of x, x as a function of y.
–1.5
–1
–0.5
1.5
1.5
1
1
1.5
1
y
y
y
0.5
0.5
0.5
0
0.5
x
1
1.5
–1.5
–1
–0.5
0
0.5
x
1
1.5
–1.5
–1
–0.5
0
–0.5
–0.5
–0.5
–1
–1
–1
–1.5
–1.5
–1.5
0.5
x
1
1.5
7. Find the value of p if the following pairs of straight lines are parallel: x6 − y2 = 1 and (p + 1)x − (2p − 3)y + 7 = 0.
[Read p.95 of the text for the conditions for two lines to be parallel and pendicular.]
8. Find the value of p if the following pairs of straight lines are perpendicular:
x−y
3
= y and (p+1)x+(p−2)y+10 = 0.
9. Find the equation of the line passing through the point (4, 2) and
a)
b)
parallel to 4x − y + 1 = 0,
perpendicular to 2x − 8y + 3 = 0.
10. Sketch the pair of straight lines in each of the following systems of linear equations. Solve each system (if possible):
a)
y =x+4
−x = y
Problem Set 1
Math 1090-001 (Spring 2001)
Due: Friday Jan. 26, 2001
b)

 2x + 3y = 6
c)

y = − 2x
3 +2
2x + 3y = 1
4x + 6y = 0
11. Solve each of the following systems of linear equations:
a)

 x + 2y + 3z = 1
3x + 3y + 4z = 3

x + 4y + 5z = 4
b)
12. Consider the linear system:

 x + 2y + 3z = 1
3x + 2y + z = 1

7x + 2y − 3z = 1

 x + y − z = −2
3x − 5y + 13z = 18

x − 2y + 5z = k
a)
For which value(s) of k does this system have one or infinitely many solutions?
b)
For each value of k you found above, how many solutions does the system have?
c)
Find all solutions for each value of k.
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