Written Homework 1: Pondering Prerequisites
MATH 160 Fall, 2012. Written homework due Wednesday, August 29, 2012.
Write your responses to the highlighted problems to on your own paper. Write your name on each
page of your work. Number each problem clearly. Each question will be scored on the basis of 10
points.
(1) According to Thomas’ Calculus (page 1) a function f from a set D to a set Y is a rule
that assigns a unique (single) element f (x) ∈ Y to each element x ∈ D. In high school you
probably used the vertical line test to determine whether a relationship is a function.
(a) Explain what the vertical line test is and how to use it.
(b) Explain why a relationship that passes the vertical line test must be a function. (In
other words, connect the vertical line test with the definition of a function.)
(c) If possible, give an example where the vertical line test cannot be used to determine
whether a relationship is a function. If there is no such example, explain why the
vertical line test can always be used. If you give such an example, explain why the
vertical line test cannot be used.
(2) (a) Show that if the graphs of two linear functions f (x) = ax + b and g(x) = cx + d both
have slope m = 1, then f og(x) = gof (x). (Recall that f og(x) means f (g(x).)
(b) Find several (at least 4) pairs of linear functions f (x) = ax + b and g(x) = cx + d
whose graphs do not have y-intercept 0 such that f og(x) = gof (x). Explain clearly
how you found these pairs of functions and how you could find more pairs of functions
with this property. If there are no such pairs of functions, explain how you know.
√
(3) (a) Consider the functions f (x) = x2 + 1 and g(x) = x − 1. Find gof (x) and simplify
the result. Use your calculator to verify your result and show/explain how you did it.
(There are several ways to do this.) (Recall that gof (x) means g(f (x).)
√
(b) Consider the functions F (x) = x2 + 9 and G(x) = x + 16. Find GoF (x) and simplify
the result. Use your calculator to verify your result and show/explain how you did it.
(There are several ways to do this.)
(4) The inverse y = g(x) of a function y = f (x) “reverses” the action of the function. The
inverse of y = f (x) can be found by interchanging the variables x and y and solving for y.
When two functions y = f (x) and y = g(x) are inverses, gof (x) = x and f og(x) = x.
1
2
(a) Find the inverse y = g(x) of the function f (x) =
(b) Show that when f (x) =
3x+5
2x−1
3x+5
2x−1 .
and g(x) is the inverse of f , gof (x) = x and f og(x) = x.
(Recall that f og(x) means f (g(x).)
(5) Aaron and LeAnn are reviewing their algebra skills before starting MATH 160. Aaron
√
solves the equation 4x + 9 = x + 3 by simplifying the left side. Here is Aaron’s solution:
√
4x + 9
√
√
4x + 9
√
4x + 3
√
4x
= x+3
= x+3
= x+3
= x
4x = x2
0 = x2 − 4x
0 = x(x − 4)
Aaron concludes that the solutions to the equation are x = 0 and x = 4.
LeAnn solves the same equation by squaring both sides. Here is her solution:
√
4x + 9 = x + 3
√
( 4x + 9)2 = (x + 3)2
4x + 9 = x2 + 9
0 = x2 − 4x
0 = x(x − 4)
LeAnn also concludes that the solutions to the equation are x = 0 and x = 4.
(a) Since they solved the equation two different ways and got the same answers, LeAnn
and Aaron are sure their answers are right. Are they right? Explain how you know.
(b) If either Aaron’s or LeAnn’s (or both) solution is wrong, explain what the error is,
correct the error, and complete the solution.
3
THE CASSETTE TAPE
(6) (PAR Problem)
--~
This diagram represents a tape recorder just as it is beginning to play a tape. The tape
Thisfrom
diagram
represents
a cassette
as itand
is beginning
playa from
tape. the
passes
the “head”
(labeled
H) at arecorder
constantjust
speed
the tape istowound
The tape passes the "head" (Labelled H) at a constant speed and the tape is
left-hand
to left
the right
beginning,
the radius of the tape on the
wound spool
fromonthe
hand hand
spool spool.
on to At
the the
right
hand spool.
left-hand spool is 2.5cm. The tape lasts 45 minutes.
At the beginning, the radius of the tape on the left hand spool is 2.5 Cill. The tape
(a) Sketch a graph to show how the length of the tape on the left-hand spool changes with
lasts 45 minutes.
time.
(i) S ketch a graph to show how the length of the tape on the left hand spool
(b) Sketch a graph to show how the radius of the tape on the left-hand spool changes with
changes with time.
time.
(c) Describe and explain how the radius on the tape of the right-hand spool changes with
time.
Length of tape
on left hand
spool
o
10
30
20
40
50
Time (minutes)
(continued)
46