CHAPTER 0 BEFORE CALCULUS SECTION 0.1...
Transcript of CHAPTER 0 BEFORE CALCULUS SECTION 0.1...
CHAPTER 0 BEFORE CALCULUS
Dr. D Page 1
SECTION 0.1 FUNCTIONS, [p1]
DEFINITIONS
1. [p2] Function
(Definition 0.1.2) If a variable y depends on a
variable x in such a way that each value of x
determines exactly one value of y, then we say that
y is a function of x.
(Definition 0.1.2)A function f is a rule that
associates a unique output with each input. y =
f(x)
Input Output
x f(x)
2. [p4] Graph of Basic Functions Graph of the function f is the graph of the equation
y = f(x)
Figure 0.1.4 [p4]
3. [p5] Vertical Line Test (Definition 0.1.3) A curve in the xy plane is the graph of some
functions f if and only if no vertical line intersects
the curve more than once.
Figure 0.1.7 [p5]
This curve cannot be the graph of a function.
4. [p5] Absolute Value Function Absolute value function is defined by
f(x) = |x| = {
Figure 0.1.9 [p6]
5. [p6] Piecewise-Defined Functions A function that consists of two or more equations.
Example is absolute value function.
6. [p7] Domain and Range Domain is the set of all possible values of x
Range is the set of all possible values of y
[p7] Natural domain of the function consists of
all real numbers for which the formula yields a real
value.
Figure 0.1.12 [p7]
The projection of y = f(x) on the x-axis is the set of
allowable x-values for f, and the projection on the
y-axis is the set of corresponding y-values.
Examples [p12]
] Use the accompanying graph to answer the
following questions, making reasonable
approximations where needed.
a. For what values of x is y = 1?
Ans: -2.9, -2, 2.35, 2.9
b. For what values of x is y = 3? Ans: none
c. For what values of y is x = 3? Ans: y = 0
d. For what values of x is ?
Ans:
e. What are the maximum and minimum values
of y and for what values of x do they occur?
f. Ans: ymax = 2.8 at x = -2.6; ymin -2.2 at x = 1.2
CHAPTER 0 BEFORE CALCULUS
Dr. D Page 2
SECTION 0.1 FUNCTIONS, [p1]
] Practice Exercise: Use the accompanying table
to answer the questions posed in
]
x -2 -1 0 2 3 4 5 6
y 5 1 -2 7 -1 1 0 9
] Find f(0), f(2), f(-2), f(3), f(√ ) and f(3t)
a. f(x) = 3x2 – 2
f(0) = 3(0)2 – 2 = 0 – 2 = -2
f(2) = 3(2)2 – 2 = 12 – 2 = 10
f(-2) = 3(-2)2 – 2 = 12 – 2 = 10
f(3) = 3(3)2 – 2 = 27 – 2 = 25
f(√ ) = 3(√ )2 – 2 = 6 – 2 = 4
f(3t) = 3( )2 – 2 = 27t
2 – 2
b. Practice Exercise: f(x) = {
] Find the natural domain and determine the range
of each function.
a. ( )
3
To determine the range, solve x in terms of y
y(x – 3) = 1
xy – 3y = 1
xy = 1 + 3y
x =
Range: 0
b. ( )
Since |x| = {
f(x) =
= {
0
Range: {1, -1}
c. ( ) √
->
√
Interval
( √ ) +
( √ √ ) -
(√ ) +
√ √ 0
Thus,
Natural Domain: √ or √ ,
Range:
d. ( ) √
Domain:
√
( )
√( ) = √
= √
√
As y varies, the value of √ varies over
the interval [0, )
√ varies over the interval in
the Range [2, ) or y
CHAPTER 0 BEFORE CALCULUS
Dr. D Page 3
SECTION 0.1 FUNCTIONS, [p1]
e. ( )
Since sin x 1,
Natural Domain: , x
For such x,
Thus, witten as
( )
( )
( )
Thus,
implies,
Range:
f. ( ) √
Division by 0 occurs for x = 2. For all other x,
, implies .
Natural Domain: or x >2
or [ ) ( )
The range of √ is [0, ), but we exclude
x = 2 for which √ =√
Range: or y > 2
or [ ) ( )
] Practice Exercise: Find the natural domain and
determine the range of each function.
a. f(x) = √ b. F(x) = √
c. g(x) = 3 + √ d. G(x) = x3 + 2
e. h(x) = 3sin x f. H(x) = ( √ )
] Use the equation y = x
2 - 6x + 8 to answer the
following questions.
a. For what values of x is y = 0?
b. For what values of x is y = -10?
c. For what values of x is ?
d. Does y have a minimum value? A maximum
value? If so, find them
Answer:
a. When y = 0, x2 - 6x + 8 = 0
(x – 2)(x – 4) = 0, implies x = 2, and 4
b. When y = -10, x2 - 6x + 8 = -10
x2 - 6x + 8 + 10 = 0
x2 - 6x + 18 = 0, no solution
Thus, x has no value
c. When , x2 - 6x + 8
(x – 2)(x – 4)
Interval x – 2 x – 4 (x – 2)(x – 4)
( ) - - +
( ) + - -
( ) + + +
2 or 4 0
Thus, x is ( or [ )
d. Maximum and Minimum value
y = x2 - 6x + 8
y – 8 = x2 – 6x
y – 8 + 9 = x2 – 6x + 9
y +1 = (x – 3)2,
V(3, -1) and graph is parabola opening up
Thus, minimum value is -1 and
no maximum value.
] Practice Exercise: Use the equation y = 1 + √
to answer the following questions.
a. For what values of x is y = 4?
b. For what values of x is y = 0?
c. For what values of x is ?
-2 2
+
-
-
-
-
-
CHAPTER 0 BEFORE CALCULUS
Dr. D Page 4
SECTION 0.2 NEW FUNCTIONS FROM OLD, [p15]
DEFINITIONS
1. [p15] Arithmetic Operations on Functions
Given functions f and g, we define
a. (f + g)(x) = f(x) + g(x)
b. (f - g)(x) = f(x) - g(x)
c. (f g)(x) = f(x) g(x)
d. (
) ( )
( )
( ) ; g(x) 0
2. [p17] Composition of Functions
(Definition 0.2.2)
Given functions f and g, the composition of f with
g, denoted by ( )( ) ( ( ))
3. [p20] Geometric Effect on Operations of Functions
Let y = f(x) be a function
a. Table 0.2.2 [p20]: Translation Principles
Operation on
y=f(x)
Add +c to f(x) Subtract +c
from f(x)
Add +c to x Subtract +c from
x
New Equation y = f(x) + c y = f(x) – c y = f(x + c) y = f(x – c)
Geometric
Effect
Translate graph
of y=f(x) c units
up
Translate graph
of y=f(x) c
units down
Translate graph of
y=f(x) c units left
Translate graph of
y=f(x) c units right
Example
b. Table 0.2.3 [p21]: Reflection Principles
Operation on y = f(x) Replace x by -x Multiply f(x) by -1
New Equation y = f(-x) y = - f(x)
Geometric Effect Reflect graph of y=f(x) about the y-axis Reflect graph of y=f(x) about the x-axis
Example
c. Fig 0.2.7 [p23]: Symmetry
𝑦 √ 𝑥 𝑦 √𝑥 𝑦 √𝑥
𝑦 √𝑥
y = x2+2
y = x2
y = x2
y = x2-2
y = (x + 2)2 y = x2 y = x2 y = (x - 2)2
CHAPTER 0 BEFORE CALCULUS
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SECTION 0.2 NEW FUNCTIONS FROM OLD
d. Table 0.2.4 [p22]: Stretching and Compressing Principles
Illustrations:
1. Figure 0.2.3 page 20
√ √ √
2. Figure 0.2.4 page 21
3. Figure 0.2.5 page 21
y = x2
y = (x-2)2
y = (x-2)2+1
𝑦 √𝑥3
𝑦 √ 𝑥3
𝑦 √ 𝑥3
CHAPTER 0 BEFORE CALCULUS
Dr. D Page 6
SECTION 0.2 NEW FUNCTIONS FROM OLD
Examples [p25] Find the formulas for f + g, f – g, fg and f/g and state
the domains of the functions.
] ( ) √ ; ( ) √
a. f + g = √ + √ = √
Domain: x
b. f – g = √ - √ = √ Domain: x
c. fg = √ (√ ) = 2(x – 1) = 2x - 2 Domain:
d.
√
√
Domain:
] Practice Exercise: ( )
; ( )
] Let ( ) √ ; ( ) , find
a. f(g(2))
( )
( )
f(g(2)) = f(9) = √ = 3
b. Practice Exercise: g(f(4))
c. f(f(16))
( ) √
( ) √ = 4
( ( )) ( ) √ = 2
d. Practice Exercise: g(g(0))
e. f(2 + h)
f(2 + h) = √
f. Practice Exercise: g(3 + h)
Find the formulas for and and state the
domains of the compositions.
] f(x) = x
2 , ( ) √
= f(g(x))
= f(√ )
= (√ )2
= 1 – x
Domain:
= g(f(x))
= g(x2)
= √
Domain: |x| , because √( )( )
Interval 1 – x 1 + x (1-x)(1+x)
( ) + - -
-1 + 0 0
( ) + + +
1 0 + 0
( ) - + -
] Practice Exercise: ( ) √ ;
( ) √
] Practice Exercise: ( )
, ( )
] ( )
; ( )
= f(g(x)) = f(
) =
(
)
=
=
=
Domain:
= g(f(x))
= g(
)
=
=
Domain:
CHAPTER 0 BEFORE CALCULUS
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SECTION 0.3 FAMILIES OF FUNCTIONS, [p29] ILLUSTRATIONS
1. [p27] Families of Curves
Family of y = mx + b Family of y = mx + b
(b fixed, m varying) (m fixed, b varying)
Fig 0.3.2
2. [p28] Power Functions: The Family of y = xn
y = x y = x2 y = x3
Fig 0.3.3
y = x4 y = x5
Fig 0.3.4 [p28]
3. [p29] The Family of y = x-n
Fig 0.3.5 [p29]
4. [p30] Power Functions with Non-integer Exponents
Fig 0.3.8
5. [p31] Polynomials
Fig 0.3.10
6. [p32]Rational Functions
Fig 0.3.11 [p32]
7. [p32] Algebraic Functions
Fig 0.3.12
CHAPTER 0 BEFORE CALCULUS
Dr. D Page 8
SECTION 0.3 FAMILIES OF FUNCTIONS, [p29]
Examples [p36]
] In each part, match the equation with one of the
accompanying graphs.
Answer: a. VI b. IV c. III d. V e. I f. II
Determine whether the statement s true or false.
(Numbers 25-27)
] Each curve in the family y = 2x + b is parallel to
the line y = 2x.
True. The graph of y = 2x + b is obtained by
translating the graph of y = 2x up b units
(or down –b units)
] Practice Exercise: Each curve in the family
y = x2 + bx +c is the translation of the graph y = x
2
] If a curve passes through the point (2, 6) and y is
inversely proportional to x, then the constant of
proportionality is 3. False, k is 12
] Find the amplitude and period, and sketch at least
two periods of the graph by hand.
a. y =3sin 4x
|a| = |3| = 3
P =
b. y = -2 cos
|a| = |-2| = 2
P =
c. y = 2 + cos
|a| = |1| = 1
P =
] Practice Exercise: Find the amplitude and
period, and sketch at least two periods of the graph by
hand.
a. y = - 1 – 4sin 2x
b. y = ½ cos ( )
c. y = - 4sin (
)
Answer: