Math 021-Z1 Summer 2014
Homework 1:
Name:
Remember to show all work for full credit.
Problem 1: Given the function, f (x) = x − x2 find the following:
a. f (2)
b. f (2 + h)
c.
f (x+h)−f (x)
h
Problem 2: Find the Domain of each function
a. f (x) =
b. f (x) =
x+2
x2 −1
√
x2 − 2x − 8 Hint: Quadratic Inequality!
Problem 3: Sketch the Graph of each Function
a. f (x) =
x2 −1
x−1


if x ≤ −1
−1,
b. f (x) = x + 2, if |x| < 1

 2
x,
if x ≥ 1
c. The line passing through the points (2, 1) and (3, 4). Also write the
equation of the line in the form y = mx + b.
Problem 4: Find the composition of functions f ◦g and g◦f for the following:
a. f (x) =
1
x
and g(x) = x3 + 2x
b. f (x) = x1/3 and g(x) = 1 −
√
x
c. Given the function F (x) = (x − 9)5 , find two functions, f and g such
that F (x) = f ◦ g
Problem 5: Solve the following equations: Hint: For parts c and d the unit
circle comes in handy!
a. 3x+1 = 9x
b. ln x + ln 3x = 0
c. cos(x) = 0 Express your answer in radians.
d. tan(x) = 1 Express your answer in radians.
e. ex
2
−1
=1
Problem 6: Use the properties of limits to help decide whether each limit
exists. If a limit exists, find its value.
x2 − 3x − 10
x→−2
x+2
a: lim
√
b: lim
x→1
8+x
x+1
x2 − 81
c: lim √
x→9
x−3
t3 − t
d: lim 2
t→1 t − 1
Problem 7: Find all values of x where the function is continuous. Write
your answer in interval notation.
√
e x−1
a: f (x) =
x−3
b: f (x) = csc(x) =
1
sin(x)
Your intervals should be expressed in radians
Problem 8: Using the properties of limits (or the ideas of horizontal and
vertical asymptotes) decide whether each infinite limit exists. If the limit
exists, find its value
√
36x2 + 2x + 7
a. lim
x→∞
3x + 2
8 − 4x3
x→∞ 5x2 + 7x + 10
b: lim
5x + 3x2
c: lim
x→∞ 6 + 7x + 4x2