Transcript Document

HCMUT – DEP. OF MATH. APPLIED
LEC 2b: BASIC ELEMENTARY
FUNCTIONS
Instructor: Dr. Nguyen Quoc Lan (October, 2007)
CONTENT
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1- POWER FUNCTION
8- HYPERBOLIC FUNCTION
2- ROOT FUNCTION
3- RATIONAL FUNCTION
4- TRIGONOMETRIC FUNCTION
5- EXPONENTIAL FUNCTION
6- LOGARITHMIC FUNCTION
7- INVERSE FUNCTION: TRIGONOMETRIC
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 left
and those of x3, x5 on the right
Root functions
(ii) a=1/n, where n is a positive integer
Then
1
n
f ( x)  x  n x is called a root function
 [0, )
domain ( f )  
(, )
if n is even
if n is odd
The graph of f is similar to that of
if n is even and
y similar
x to that of
if n is odd
y3 x
(1,1)
f ( x)  x
(1,1)
f ( x)  3 x
(iii) When a=–1 ,
1 reciprocal function
is
the
f ( x)  x 
x
1
The graph is a hyperbola with the coordinate axes as its asymptotes
Rational functions
A rational function is the ratio of two polynomials:
P( x)
f ( x) 
Q( x)
Where P and Q are polynomials. The domain of f consists of all real
number x such that Q(x)  0.
1
f ( x) 
x
{x/x  0}
is a rational function whose domain is
2 x 4  x 2  1 Domain(f)={x/ x  2}
f ( x) 
x2  4
Trigonometric functions
f(x)=sinx
g(x)=cosx
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]
Exponential functions
These are functions of the form f(x)=ax, a > 0
y=2x
y=(0.5)x
Logarithmic functions
These are functions f(x)=logax, a > 0. They are inverse of exponential
functions
log2x
log3x
log10x
log5x
Definition. A function f is a one-to-one function if:
x1  x2  f(x1)  f(x2)
4
3
10
7
4
2
2
1
f is one-to-one
f
4
3
2
1
10
4
g
2
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 = x23 then
(x1 – x2)(x12+ x1x2+ x22) = 0  x1 = x2 because
1
1 2
2
x  x1 x2  x  ( x1  x2 )  ( x1  x22 )  0
2
2
2
1
2
2
hence f(x) = x3 is one-to-one
Inverse functions
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:
f -1(y) = x  f(x) = y, for all y in B
domain( f –1) = range (f)
range(f -1) = domain(f)
Example. Let f be the following function
4
-10
3
7
1
3
f
A
B
Then f -1 just reverses the effect of f
4
-10
3
7
1
3
f -1
A
B
If we reverse to the independent variable x then:
f -1(x) = y  f(y) = x, for all x in B
f -1(f(x)) = x, for all x in A
f(f -1(x)) = 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:
x3  y  2
x  3 y2
Interchange x and y:
1
y  x  2  f ( x)
3
Inverse trigonometric functions
Question: When the trigonometric funtion y = sinx is one – to – one and
how about its inverse function?
 

x    ,  , y   1,1 : sin x  y  x  arcsin y
 2 2
 

Inverse function : y  arcsin x, x   1,1, y    ,   x  sin y
 2 2
Application: Compute the integral

dx
1 x2
Inverse trigonometric functions
Considering analogicaly for the functions y = cosx, y = tgx, y = cotgx, we
give the definition of three others inverse trigonometric functions
y  arccos x, x   1,1, y  0,    x  cos y
 

y  arctanx, x  R, y    ,   x  tany
 2 2
y  arc cot x, x  R, y  0,    x  cot y
Application: Compute the integral
dx
 1 x2
Hyperbolic functions
The four next functions are called hyperbolic function
e x  ex
sinh x  sh x 
2
e x  ex
cosh x  chx 
2
sh x e x  e  x
tanh x  thx 
 x
chx e  e  x
chx
1
coth x 

sh x thx
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

dx
1 x2
Piecewise defined functions
1  x if x  1
f ( x)   2
 x if x  1
f(0)=1-0=1, f(1)=1-1=0
1
1
and f(2)=22=4
The graph consists of half a line with slope –1 and y-intercept 1;
and part of the parabola y = x2 starting at the points
(excluded)
(1,1)