Math 211-330/332 Worksheet 1 Jan 20, 2016 Solutions Review Math 211-330/332 Worksheet 1
Jan 20, 2016
Solutions
Review:
• Definitions: function, domain, range, independent variable, dependent variable;
• Think of a function as a machine (I prefer “vending machine”);
• Representations of functions: words, table, graph, mathematical formula;
• How to determine the domain of a function;
• The vertical line test.
1. Some of these algebraic manipulations are correct, and others are incorrect. (Note
that a correct answer might be obtained by using incorrect reasoning.) Identify the
correct ones and fix the incorrect ones:
(a)
(x + y)2
x2 + y 2
=
= x2 + y.
y
y
Solution. This is incorrect.
(x + y)2
x2 + 2xy + y 2
x2
Correct version:
=
=
+ 2x + y.
y
y
y
&Ccedil;
&aring;2
&Aring; &atilde;2
x2
1
(x2 )3
2
=
x
&middot;
= x2 &middot;
(b)
3
2
3
(x )
x
x
Solution. This is correct.
(c)
=
x2
= 1.
x2
x2 1
x2 − 1
=
− = x − 1.
x+1
x
1
Solution. This is incorrect.
x2 − 1
(x − 1)(x + 1)
Correct version:
=
= x − 1.
x+1
x+1
&Aring;
&atilde;
2−4
−4 −2
4
1
(d) −2 =
=
= .
2
−2
16
4
Solution. This is incorrect.
2−4
22
1
1
Correct version: −2 = 4 = 2 = .
2
2
2
4
&Aring;
(e)
x−1 + y −1
(x + y)−1
x+y
=
=
−1
−1
x −y
(x − y)−1
x−y
Solution. This is incorrect.
Correct version:
&atilde;−1
=−
x+y
x+y
=
.
x−y
x−y
y+x
1 1
+
y+x
x−1 + y −1
x y
xy
= y−x =
=
.
−1
−1
1 1
x −y
y−x
−
xy
x y
2. Determine the domain of each function
√
(a) f (x) = x2 + 1
(b) f (x) =
1
x2 −3x+2
Solution. (a) x2 + 1 ≥ 0 ⇒ x ∈ (−∞, +∞);
(b) x2 − 3x + 2 6= 0 ⇒ (x − 1)(x − 2) 6= 0 ⇒ x 6= 1, 2
3. If f (x) = x2 − 3x + 2, evaluate
(a) f (1)
(b) f (t)
(c) f (x + 1)
(d)
f (1+t)−f (1)
t
Solution. (a) f (1) = 12 − 3 ∗ 1 + 2 = 0
(b) f (t) = t2 − 3t + 2
(c) f (x + 1) = (x + 1)2 − 3(x + 1) + 2
(d)
f (1+t)−f (1)
t
=
[(1+t)2 −3(1+t)+2]−0
t
=
1+2t+t2 −3−3t+2
t
=
t2 −t
t
=
t(t−1)
t
=t−1