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HCMUT – DEP. OF MATH. APPLIED LEC 2b: BASIC ELEMENTARY FUNCTIONS Instructor: Dr. Nguyen Quoc Lan (October, 2007)

HCMUT – DEP. OF MATH. APPLIED LEC 2b: BASIC ELEMENTARY FUNCTIONS Instructor: Dr. Nguyen Quoc Lan (October, 2007)

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Page 1: HCMUT – DEP. OF MATH. APPLIED LEC 2b: BASIC ELEMENTARY FUNCTIONS Instructor: Dr. Nguyen Quoc Lan (October, 2007)

HCMUT – DEP. OF MATH. APPLIED

LEC 2b: BASIC ELEMENTARY FUNCTIONS

Instructor: Dr. Nguyen Quoc Lan (October, 2007)

Page 2: HCMUT – DEP. OF MATH. APPLIED LEC 2b: BASIC ELEMENTARY FUNCTIONS Instructor: Dr. Nguyen Quoc Lan (October, 2007)

CONTENT--------------------------------------------------------------------------------------------------------------------------------

-

1- POWER

FUNCTION 2- ROOT

FUNCTION 3- RATIONAL

FUNCTION 4- TRIGONOMETRIC

FUNCTION 5- EXPONENTIAL

FUNCTION 6- LOGARITHMIC FUNCTION

7- INVERSE FUNCTION:

TRIGONOMETRIC

8- HYPERBOLIC

FUNCTION

Page 3: HCMUT – DEP. OF MATH. APPLIED LEC 2b: BASIC ELEMENTARY FUNCTIONS Instructor: Dr. Nguyen Quoc Lan (October, 2007)

Power Function

The function y=xa , where a is a constant is called a power function

(i) When a=n, a positive integer, the graph of f is similar to the parabola y=x2 if n is even and similar to the graph of y=x3 if n is odd

However as n increases, the graph becomes flatter near 0 and steeper when x 1

Page 4: HCMUT – DEP. OF MATH. APPLIED LEC 2b: BASIC ELEMENTARY FUNCTIONS Instructor: Dr. Nguyen Quoc Lan (October, 2007)

The graphs of x2, x4, x6 on the leftand those of x3, x5 on the right

Page 5: HCMUT – DEP. OF MATH. APPLIED LEC 2b: BASIC ELEMENTARY FUNCTIONS Instructor: Dr. Nguyen Quoc Lan (October, 2007)

(ii) a=1/n, where n is a positive integer

Then is called a root function

nn xxxf 1

)(

xy 3 xy

),(

),0[)( fdomain

Root functions

if n is even

if n is odd

The graph of f is similar to that of if n is even and similar to that of if n is odd

Page 6: HCMUT – DEP. OF MATH. APPLIED LEC 2b: BASIC ELEMENTARY FUNCTIONS Instructor: Dr. Nguyen Quoc Lan (October, 2007)

xxf )( 3)( xxf

(1,1) (1,1)

Page 7: HCMUT – DEP. OF MATH. APPLIED LEC 2b: BASIC ELEMENTARY FUNCTIONS Instructor: Dr. Nguyen Quoc Lan (October, 2007)

(iii) When a=–1 , is the reciprocal function x

xxf1

)( 1

The graph is a hyperbola with the coordinate axes as its asymptotes

Page 8: HCMUT – DEP. OF MATH. APPLIED LEC 2b: BASIC ELEMENTARY FUNCTIONS Instructor: Dr. Nguyen Quoc Lan (October, 2007)

Rational functions

A rational function is the ratio of two polynomials:

)(

)()(

xQ

xPxf

xxf

1)( is a rational function whose

domain is

{x/x 0}

Where P and Q are polynomials. The domain of f consists of all real number x such that Q(x) 0.

Page 9: HCMUT – DEP. OF MATH. APPLIED LEC 2b: BASIC ELEMENTARY FUNCTIONS Instructor: Dr. Nguyen Quoc Lan (October, 2007)

4

12)(

2

24

x

xxxf Domain(f)={x/ x 2}

Page 10: HCMUT – DEP. OF MATH. APPLIED LEC 2b: BASIC ELEMENTARY FUNCTIONS Instructor: Dr. Nguyen Quoc Lan (October, 2007)

Trigonometric functions

sinx and cosx are periodic functions with period 2 : sin(x + 2 ) = sinx, cos(x + 2 ) = cosx, for every x in R the domains of sinx and cosx are R, and their ranges are [-1,1]

f(x)=sinx g(x)=cosx

Page 11: HCMUT – DEP. OF MATH. APPLIED LEC 2b: BASIC ELEMENTARY FUNCTIONS Instructor: Dr. Nguyen Quoc Lan (October, 2007)

These are functions of the form f(x)=ax, a > 0

Exponential functions

y=2x y=(0.5)x

Page 12: HCMUT – DEP. OF MATH. APPLIED LEC 2b: BASIC ELEMENTARY FUNCTIONS Instructor: Dr. Nguyen Quoc Lan (October, 2007)

Logarithmic functions

These are functions f(x)=logax, a > 0. They are inverse of exponential functions

log2x log3x

log10

xlog5x

Page 13: HCMUT – DEP. OF MATH. APPLIED LEC 2b: BASIC ELEMENTARY FUNCTIONS Instructor: Dr. Nguyen Quoc Lan (October, 2007)

Definition. A function f is a one-to-one function if:

x1 x2 f(x1) f(x2)

43

2

1

43

2

1

107

4

2

104

2

f

g

f is one-to-one

g is not one-to-one :

2 3 but g(2) = g(3)

Page 14: HCMUT – DEP. OF MATH. APPLIED LEC 2b: BASIC ELEMENTARY FUNCTIONS Instructor: Dr. Nguyen Quoc Lan (October, 2007)

Example. Is the function f(x) = x3 one-to-one ?

Solution1. If x13 = x2

3 then

(x1 – x2)(x12+ x1x2+ x2

2) = 0 x1 = x2 because

0)(2

1)(

2

1 22

21

221

2221

21 xxxxxxxx

hence f(x) = x3 is one-to-one

Page 15: HCMUT – DEP. OF MATH. APPLIED LEC 2b: BASIC ELEMENTARY FUNCTIONS Instructor: Dr. Nguyen Quoc Lan (October, 2007)

Definition. Let f be a one-to-one function with domain A and range B. Then the inverse function f -1 has domain B and range A and is defined by:

domain( f –1) = range (f)

range(f -1) = domain(f)

f -1(y) = x f(x) = y, for all y in B

Inverse functions

Page 16: HCMUT – DEP. OF MATH. APPLIED LEC 2b: BASIC ELEMENTARY FUNCTIONS Instructor: Dr. Nguyen Quoc Lan (October, 2007)

4

3

1

-10

7

3

f

Example. Let f be the following function

A B

Page 17: HCMUT – DEP. OF MATH. APPLIED LEC 2b: BASIC ELEMENTARY FUNCTIONS Instructor: Dr. Nguyen Quoc Lan (October, 2007)

4

3

1

-10

7

3

f -1

Then f -1 just reverses the effect of f

A B

Page 18: HCMUT – DEP. OF MATH. APPLIED LEC 2b: BASIC ELEMENTARY FUNCTIONS Instructor: Dr. Nguyen Quoc Lan (October, 2007)

f -1(f(x)) = x, for all x in A

f(f -1(x)) = x, for all x in B

If we reverse to the independent variable x then:

f -1(x) = y f(y) = x, for all x in B

How to find f –1

Step1 Write y = f(x)

Step2 Solve this equation for x in terms of y

Step3 Interchange x and y.

The resulting equation is y = f -1(x)

Page 19: HCMUT – DEP. OF MATH. APPLIED LEC 2b: BASIC ELEMENTARY FUNCTIONS Instructor: Dr. Nguyen Quoc Lan (October, 2007)

Example. Find the inverse function of f(x) = x3 + 2

Solution. First write y = x3 + 2

Then solve this equation for x:

3

3

2

2

yx

yx

Interchange x and y:

)(2 13 xfxy

Page 20: HCMUT – DEP. OF MATH. APPLIED LEC 2b: BASIC ELEMENTARY FUNCTIONS Instructor: Dr. Nguyen Quoc Lan (October, 2007)

Question: When the trigonometric funtion y = sinx is one – to – one and how about its inverse function?

Inverse trigonometric functions

yxyxyx arcsinsin:1,1,2

,2

yxyxyx arcsinsin:1,1,

2,

2

yxyxxy sin2

,2

,1,1,arcsin :function Inverse

yxyxxy sin

2,

2,1,1,arcsin :function Inverse

Application: Compute the integral

21 x

dx

Page 21: HCMUT – DEP. OF MATH. APPLIED LEC 2b: BASIC ELEMENTARY FUNCTIONS Instructor: Dr. Nguyen Quoc Lan (October, 2007)

Considering analogicaly for the functions y = cosx, y = tgx, y = cotgx, we give the definition of three others inverse trigonometric functions

Inverse trigonometric functions

yxyxxy cos,0,1,1,arccos

yxyRxxy tan2

,2

,,arctan

yxyRxxy cot,0,,cotarc

yxyxxy cos,0,1,1,arccos

yxyRxxy tan2

,2

,,arctan

yxyRxxy cot,0,,cotarc

Application: Compute the integral

21 xdx

Page 22: HCMUT – DEP. OF MATH. APPLIED LEC 2b: BASIC ELEMENTARY FUNCTIONS Instructor: Dr. Nguyen Quoc Lan (October, 2007)

The four next functions are called hyperbolic function

Hyperbolic functions

2shsinh

xx eexx

2

chcoshxx ee

xx

xx

xx

ee

eexx

xx

chsh

thtanhxx

xx

th1

shch

coth

2shsinh

xx eexx

2

chcoshxx ee

xx

xx

xx

ee

eexx

xx

chsh

thtanhxx

xx

th1

shch

coth

We get directly hyperbolic formulas from all familiar trigonometric formulas by changing cosx to coshx and sinx to isinhx (i: imaginary number, i2 = –1)

Page 23: HCMUT – DEP. OF MATH. APPLIED LEC 2b: BASIC ELEMENTARY FUNCTIONS Instructor: Dr. Nguyen Quoc Lan (October, 2007)

Hyperbolic formulas

Application: Compute the integral

21 x

dx

Page 24: HCMUT – DEP. OF MATH. APPLIED LEC 2b: BASIC ELEMENTARY FUNCTIONS Instructor: Dr. Nguyen Quoc Lan (October, 2007)

Piecewise defined functions

1

1

1

11)( 2 xifx

xifxxf

f(0)=1-0=1, f(1)=1-1=0

and f(2)=22=4The graph consists of half a line with slope –1 and y-intercept 1; and part of the parabola y = x2 starting at the points (1,1) (excluded)