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Lê Xuân Đại HCMUT-OISP FUNCTIONS OF SINGLE VARIABLE HCMC — 2016... Lê Xuân Đại HCMUT-OISP FUNCTIONS OF SINGLE VARIABLE HCMC — 2016... Lê Xuân Đại HCMUT-OISP FUNCTIONS OF SINGLE VARIABLE

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FUNCTIONS OF SINGLE VARIABLE

E LECTRONIC VERSION OF LECTURE

Dr Lê Xuân Đại

HoChiMinh City University of Technology Faculty of Applied Science, Department of Applied Mathematics

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1 F UNCTIONS OF SINGLE VARIABLE

2 B ASIC PROPERTIES OF FUNCTIONS

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A function f is a rule that assigns to each element x in

a set X ⊂ R exactly one element y , called f (x) in a set

The set X = {x ∈ R : f (x) is defined}is called the domain

of the function f and is denoted by D(f ). The set

function f and is denoted by E(f ).

f (x)is the value off atxand is read "f of x"

Dr Lê Xuân Đại (HCMUT-OISP) FUNCTIONS OF SINGLE VARIABLE HCMC — 2016 3 / 52

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The set consists of all points (x, f (x)), x ∈ X in the coordinate plane Oxy is called the graph of the function f

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Find the domain and range of function f (x) =px + 2

SOLUTION

1 The domain off consists of all values ofxsuch

interval[−2,+∞)

2 The range off consists of all values ofy such that

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The functions which are defined by different formulas

in different parts of their domain, are called

piecewise defined functions

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Suppose thaty = f (u),whereuis a function ofx :

Given 2 functions f and g, the composite function

(f ◦ g)(x) = f (g(x))

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Let f be a one-to-one function with domain D and range E. Then its inverse function f−1 (read: f inverse) has domain E and range D and is defined by

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HOW TO FIND THE INVERSE FUNCTION OF A ONE-TO-ONE FUNCTIONf

1 Writey = f (x)

2 Solve this equation forx in terms ofy (if possible)

3 To expressf−1as a function ofx,interchangex

andy The resulting equation isy = f−1(x)

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The function f is called periodic of period T > 0 if for

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1 Function f (x) = x is increasing on theR.

2 Function g(x) = x2 is decreasing on the interval

(−∞,0)and increasing on the interval(0, +∞).

3 Function h(x) = c = const, according to the definition is not decreasing and not increasing.

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Let function f : X → Y be defined on a set D ⊂ X.

Function f is called

1 bounded from above if there is a number M ∈ R

such that for all x ∈ D from the domain D one has

f (x) É M;

2 bounded from below if there is a number m ∈ R

such that for all x ∈ D from the domain D one has

f (x) Ê m;

3 bounded if there is a number C > 0 such that for

4 unbounded if for all C > 0, exists x0∈ D such that

|f (x0)| > C.

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LINEAR FUNCTION

mis the slope of the line andbis they−intercept.The slope,m, of the line through(x1, y1)and(x2, y2)isgiven by the following equation, ifx16= x2

m = y2− y1

x2− x1The slope of a line can be interpreted as the rate ofchange in they−coordinates for each 1-unit increase

in the x−coordinates

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A function P is called a polynomial if

P(x) = a n x n + a n−1 x n−1 + + a2x2+ a1x + a0

where n is a nonnegative integer and the numbers

a0, a1, a2, , a n are constants called the coefficients of the polynomial.

If the leading coefficienta n6= 0,then thedegreeofthe polynomial isn

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degree3- cubic function

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Caseα = 2 ⇒ y = x2 - square function

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Caseα =1

2 ⇒ y =px- square root function

1 Domain:D = [0,+∞)

2 Range:E = [0,+∞)

3 Function is increasing on the interval(0, +∞)

4 Function does not have symmetry

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a x .b x = (ab) x

a x

b x =

³a b

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Function y = a x , (a > 1)

1 Domain:D = R

2 Range:E = (0,∞)

3 Function is increasing on the interval(−∞,+∞)

4 The graph always passes through the point at

(0, 1)

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Function y = a x , (0 < a < 1)

1 Domain:D = R

2 Range:E = (0,+∞)

3 Function is decreasing on the interval(−∞,+∞)

4 The graph always passes through the point at

(0, 1)

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LOGARITHMIC FUNCTIONS

(Read: logarithm ofxwith basea)Logarithmic functiony = log a xis the inverse function

of exponential functiony = a x,this means that, if

logarithmic function is:

D = {x ∈ R | x > 0}.

The graph of logarithmic functiony = log a xis thereflection of the graph of exponential functiony = a x

about the line y =x

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Properties of logarithmic functions

log a (x.y) = log a x + log a y log a

µx y

¶

= log a x − log a y

log aµ 1x

¶

= −log a x log a β x α= α

β log a x

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Function y = log a x, (a > 1)

1 Domain:D = (0,+∞)

2 Range:E = R

3 Function is increasing on the interval(0, +∞)

4 The graph of logarithmic functiony = log a xis thereflection of the graph of exponential function

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Function y = log a x, (0 < a < 1)

1 Domain:D = (0,+∞)

2 Range:E = R

3 Function is decreasing on the interval(0, +∞)

4 The graph of logarithmic functiony = log a xis thereflection of the graph of exponential function

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Function siney = sinx

1 Domain:D = R 2 Range:E = [−1,1]

3 Function is periodic of period2π :

sin(x) = sin(x + 2π) = sin(x − 2π)

4 Function is increasing on the interval¡−π2,π2¢ ,anddecreasing on the interval¡π

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Function cosiney = cosx

1 Domain:D = R 2 Range:E = [−1,1]

3 Function is periodic of period2π :

cos(x) = cos(x + 2π) = cos(x − 2π)

4 Function is increasing on the interval(−π,0),anddecreasing on the interval(0,π),

5 Function is even, the graph is symmetric withrespect to they−axis

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Function tangenty = tanx = cos x sin x

2 Range:E = R

3 Function is periodic of periodπ :

tan(x) = tan(x + π) = tan(x − π)

4 Function is increasing on the interval¡−π2,π2¢

5 Function is odd, the graph is symmetric aboutthe originO(0, 0)

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Function cotangenty = cotx = cos x sin x

1 Domain:D = R \ {kπ,k ∈ Z}

2 Range:E = R

3 Function is periodic of periodπ :

cot(x) = cot(x + π) = cot(x − π)

4 Function is decreasing on the interval(0,π)

5 Function is odd, the graph is symmetric aboutthe originO(0, 0)

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Some basic formulas

2 = 1; sin(kπ) = 0 cos 2x = cos2x − sin2x

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Function arcsiney = arcsinx

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Function arcosiney = arccosx

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Function arctangenty = arctanx

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4 Function is even, thegraph is symmetricwith respect to the

y−axis

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Function tanh x = sinh x

cosh x is called hyperbolic tangent.

Function coth x = cosh x

sinh x is called hyperbolic cotangent.

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Hyperbolic Identities

sinh(−x) = −sinhx cosh(−x) = coshx

cosh2x − sinh2x = 1

sinh(x + y) = sinhx coshy + coshx sinhy cosh(x + y) = coshx coshy + sinhx sinhy

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MATLAB: FUNCTIONS

Example: syms x;f = xˆ2; g = exp(x);compose(f ,g)

⇒ ans = exp(2 ∗ x).

2 Inverse function: finverse(f ).

Example: syms x; f=exp(x);finverse(f ) ⇒ ans = log(x).

3 Evaluate function at a number: subs(f,x,a).

Example: syms x;f = xˆ2 + 1; subs(f ,x,2) ⇒ ans = 5.

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MATLAB: MATHEMATICAL EXPRESSION

1 Simplify expression: simplify(f ).

Example: syms x;f = (sin(x))ˆ2 + (cos(x))ˆ2;

simplify(f ) ⇒ ans = 1.

2 Simple expression: simple(f ).

Example: syms x;f = (x + 1) ∗ x ∗ (x − 1); simple(f ) ⇒ ans = xˆ3 − x.

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MATLAB: INPUT - OUTPUT

1 Input: input(’Input x’, x)

2 Output: disp(’The value of x is’, x)

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THANK YOU FOR YOUR ATTENTION

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