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

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

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

(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

xxf )( 3)( xxf

(1,1) (1,1)

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

xxf1

)( 1

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

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.

4

12)(

2

24

x

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

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

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

Exponential functions

y=2x y=(0.5)x

Logarithmic functions

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

log2x log3x

log10

xlog5x

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)

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

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

4

3

1

-10

7

3

f

Example. Let f be the following function

A B

4

3

1

-10

7

3

f -1

Then f -1 just reverses the effect of f

A B

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)

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

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

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

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)

Hyperbolic formulas

Application: Compute the integral

21 x

dx

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)