1.6
1.9
2.2
2.3
Section 2.2: Intro. to Function
Vocab.
function:
domain:
range:
independent variable:
dependent variable:
one-to-one function:
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2.6
Q: How does the vertical line test relate to the definition of
a function?
Q: Is there a similar way to check if a function is 1-1?
Ex. 1: Graphical Representation
(Is it a fn.? Is it 1-1? What is the domain? What is the range?)
(a)
(b)
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Ex. 1 cont.
(Is it a fn.? Is it 1-1? What is the domain? What is the range?)
(c)
(d) Given sets X = {1, 2, 3, 4, 5}
and Y = {−1, 0, 2, 3}.
f (1) = −1, f (2) = 0, f (3) = −1,
f (4) = 0, and f (5) = 3
Is f : X → Y ? Is it 1-1?
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2.3
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Ex. 2 Algebraic Representation
Find mathematical fn., and state (implied) domain.
”For each number x in the domain, the corresponding range value
y is found by adding one to the domain value and then dividing
that result into five added to five times the domain value.”
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Typical Types of Functions
Polynomial (of degree n):
f (x) = an x n + an−1 x n−1 + ... + a1 x + a0
Linear (n = 1):
f (x) = ax + b with a �= 0
Quadratic (n = 2):
f (x) = ax 2 + bx + c with a �= 0
Cubic (n = 3):
f (x) = ax 3 + bx 3 + cx + d with a �= 0
p(x)
Rational: f (x) = d(x)
, with x such that d(x) �= 0 and p(x)
and d(x) are polynomials
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Types of Functions cont.
Power fn.: f (x) = x r
integral powers: r �Z+
reciprocal powers: r �Z−
roots: r = mn where m�Z+ , n�Z+ and n �= 0
√
√
f (x) = n x m = ( n x)m
Exponential: f (x) = b x , where b > 0
Logarithmic: f (x) = logb x where b > 0
Trigonometric: examples f (x) = sinx and g (x) = secx
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2.3
Typical Domain Restrictions
Fractions
Roots
odd roots:
even roots:
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2.4
Ex. 3
Let f (x) = 3x + 2 and g (x) = 2x 2 − 1.
Find
(a) f (1)
√
(b) g ( 3)
(c) f (t 2 − 3t) − g (t + 2)
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2.3
Definition (Difference Quotient)
For a given function, f (x),
f (x+h)−f (x)
h
Geometrically -
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2.4
Ex. 4
Let f (x) = 3x + 2 and g (x) = 2x 2 − 1.
(a) Find the difference quotient of f (x).
(b) Find the difference quotient of g (x).
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2.3
Ex. 5: True or False? Explain.
(a) f (x − 1) = f (x) − 1
(b) f (x + h) = f (x) + h
(c)
f (x+h)
f (x)
=h
(d) f (3x) = 3f (x)
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Ex. 6:
Let R(x) = 3x 2 + 3x −2 − x − x −1 .
Show R( x1 ) = R(x).
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