chapter 0 before calculus section 0.1...

CHAPTER 0 BEFORE CALCULUS

Dr. D

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


Dr. D


] 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


Dr. D


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.


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.


b. For what values of x is y = 0?

c. For what values of x is ?

-2 2

+

-

-

-

-

-


Dr. D

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


Dr. D

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

√ √ √



y = x2

y = (x-2)2

y = (x-2)2+1

𝑦 √𝑥3

𝑦 √ 𝑥3

𝑦 √ 𝑥3


Dr. D

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:


Dr. D

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


Dr. D

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:

chapter 0 before calculus section 0.1...

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