hcmut – dep. of math. applied lec 2b: basic elementary functions instructor: dr. nguyen quoc lan...
TRANSCRIPT
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HCMUT – DEP. OF MATH. APPLIED
LEC 2b: BASIC ELEMENTARY FUNCTIONS
Instructor: Dr. Nguyen Quoc Lan (October, 2007)
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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
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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
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The graphs of x2, x4, x6 on the leftand those of x3, x5 on the right
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(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
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xxf )( 3)( xxf
(1,1) (1,1)
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(iii) When a=–1 , is the reciprocal function x
xxf1
)( 1
The graph is a hyperbola with the coordinate axes as its asymptotes
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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.
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4
12)(
2
24
x
xxxf Domain(f)={x/ x 2}
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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
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These are functions of the form f(x)=ax, a > 0
Exponential functions
y=2x y=(0.5)x
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Logarithmic functions
These are functions f(x)=logax, a > 0. They are inverse of exponential functions
log2x log3x
log10
xlog5x
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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)
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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
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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
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4
3
1
-10
7
3
f
Example. Let f be the following function
A B
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4
3
1
-10
7
3
f -1
Then f -1 just reverses the effect of f
A B
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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)
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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
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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
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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
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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)
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Hyperbolic formulas
Application: Compute the integral
21 x
dx
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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)