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**Unformatted text preview: **− = = =
= 3x − 1
(6x2 − 2x) x simplify complex fractions 58 Relations and Functions
3x − 1
− 1) factor = X
$1
(3x $$
$$− 1)
$
2x2$$− $
(3x $ 1) cancel = 1
2x2 = 2x2 (3x To ﬁnd the domain, we consider the ﬁrst step after substitution:
1
x
6x2 − 2x
3− 1
To avoid division by zero in the ‘little’ fraction, x , we need x = 0. For the ‘big’ fraction we
2 − 2x = 0 and solve: 2x(3x − 1) = 0 and get x = 0, 1 . Thus we must exclude x = 1 as
set 6x
3
3
well, resulting in a domain of (−∞, 0) ∪ 0, 1 ∪ 1 , ∞ .
3
3 We close this section with concept of the diﬀerence quotient of a function. It is a critical tool
for Calculus and also a great way to practice function notation.2
Definition 1.6. Given a function, f , the diﬀerence quotient of f is the expression:
f (x + h) − f (x)
h
Example 1.6.2. Find and simplify the diﬀerence quotients for the following functions
1. f (x) = x2 − x − 2 2. g (x) = 3
2x + 1 Solution.
1. To ﬁnd f (x + h), we replace every occurrence of x in the formula f (x)...

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