Calculus 1 - MTHG002 Lecture Notes (Fall 2024) PDF

Calculus 1 – MTHG002 – Fall 2024 Dr.Fayad Galal Course Description : (1) Functions, Essential Functions (Linear, Power, Polynomial, Rational, Algebraic, Trigonometric). (2) New Functions from Old Functions (Graphing, Combined, Composite Functions). (3) Exponential Functio...

Calculus 1 – MTHG002 – Fall 2024 Dr.Fayad Galal Course Description : (1) Functions, Essential Functions (Linear, Power, Polynomial, Rational, Algebraic, Trigonometric). (2) New Functions from Old Functions (Graphing, Combined, Composite Functions). (3) Exponential Functions, Inverse Functions, Logarithmic and Inverse Trigonometric Functions. (4) Limits, One Sided Limits, Limit Laws, Squeeze Theorem. (5) Limits Involving Infinity, Asymptotes of Graphs, Continuity. (6) Differentiation, Differentiation Rules, Derivatives of Power and Trigonometric Functions. (7) Chain Rule, Implicit Differentiation, Higher Order Derivatives, Parametric Differentiation. (8) Derivatives of Exponential, Inverse Trigonometric and Logarithmic Differentiation. (9) Hyperbolic and Inverse Hyperbolic Functions and Their Derivatives. (10) Indeterminate Forms and L'Hopital's Rule. (11) Taylor and Maclaurin Series. (12) Functions of Several Variables, Partial Derivatives and Their Applications. References : (1) Thomas, '' Calculus Early Transcendentals '', Pearson Education Limited, 12th Edition , 2014. (2) James Stewart, '' Calculus Early Transcendentals '', Cengage Learning, 8th Edition , 2016. 1  Calculus 1 – MTHG002 – Fall 2024 Lecture Notes (1) – Dr. Fayad Galal Precalculus Review Sets of Numbers : Natural numbers   1 , 2 , 3 , 4 ,.... Integers   ...,  2 ,  1 ,0 , 1 , 2 ,.... p  Rational numbers  p , q  ; q  0 . q  Irrational numbers   ..., 2 , 3  5 ,  , e ,.... Real numbers     . Intervals : Notation Set description Picture  a ,b  x a  x  b a ,b  x a  x  b a ,b  x a  x  b  a ,  x x  a    ,b  x x  b   ,     x    x   2 Rules for Inequalities : If a , b and c are real numbers , then (i) If a  b , then a  c  b  c. (ii) If a  b and c  0 , then ac  b c. (iii) If a  b and c  0 , then ac  b c. 1 1 (iv) If 0  a  b , then . a b Absolute Value Properties : If a , b are real numbers , then  a , a 0 (i) a  a  2.  a , a 0 (ii)  a  a. (iii) a b  a b. a a (iv) . b b Absolute Values and Intervals : If a is any positive real number , then (i) x  a  x a  x   a , a. (ii) x  a   a x a  x  a ,a . (iii) x  a   a  x  x  a  x     ,  a   a ,   or  x      a ,a . 3 Example (1) : Solve the following inequalities : (i) 3  2  x  5. (ii) 4   x  1  2  9. Solution (i) 3  2  x  5  1  x 3  1x 9. * x  1 , 9 . (ii) 4   x  1  2  9  2  x 1  3. (1) 2  x  1  (2) x  1  3 and  2  x 1  x 1  2  3  x 1  3 or 1x  x 3  2  x 4. or x    2 ,  1    3 ,4  *. x    2 ,4     1 , 3  Example (2) : Solve the following inequalities : 3 (i) x 2  x  2. (ii) 1. x 2 Solution 4 (i) x 2  x  2  x2  x  2 0   x  1  x  2   0. Sign of  x  1  Sign of  x  2  Sign of  x  1  x  2  x    ,  1   2 ,  *. x     1 ,2  3 (ii) 1 x 2 3 3 x 2 x 1  1 0  0  0. x 2 x 2 x 2 Sign of  x  1  Sign of  x  2   x 1  Sign of    x 2  * x  1 ,2 . 5  Chapter (1) : Functions 1.1 Functions and Their Graphs : Definition : A function f from a set D to a set Y is a rule that assigns a unique (single) element y  f  x  Y to each element x  D. Arrow diagram for f. Machine diagram for f. * The domain of the function is the set D of all possible input values of x. * The range of the function is the set of all values of y  f  x  as x varies through D. * The element x is called independent variable and y is called dependent variable. 6 Remark : * The graph of a function f is the graph of the equation y  f  x  for x in the domain of f. The Vertical Line Test : The graph of a function intersects any vertical line at most once. (i) This curve represents (ii) This curve doesn’t represent a function. a function. 7 Remarks : (i) x 2  y 2  a 2. (ii) y  a 2  x 2. (iii) y   a2  x 2. Circle. Upper semi-circle. Lower semi-circle. Not a function. Domain    a , a . Domain    a , a . Range  0 , a . Range    a ,0 . Functions. 8 Piecewise-Defined Functions : A piecewise-defined function is a function defined by different formulas in different parts of its domain. * The absolute value function  x , x 0 y  x 2  x   x , x 0 Domain   , Range  0 , . Example (1) : Sketch the graph of the function f defined as follows : 2 x  3 , x 0  f x    x 2 , 0 x 2 ,  1 , x 2  then find the domain and the range of the function. Solution * Domain  . * Range     , 4 . 9 Symmetry : (i) If f   x   f  x  , then f is an even function and its graph is symmetric about the y-axis. (ii) If f   x    f  x  , then f is an odd function and its graph is symmetric about the origin. Even functions. Odd functions. 10 Example (2) : Study the symmetry of the following functions : x (i) f  x   x x  2 x 3  x 5. (ii) g  x    3 x4  x6. x2 x  x4 (iii) h  x  . x 1 2 Solution (i) f  x   x x  2 x 3  x 5. * Since f   x     x   x  2   x 3    x 5   x x  2x3  x5    x x  2 x3  x5   f x  ,  then f is an odd function and its graph is symmetric about the origin. x (ii) g  x    3 x4  x6. x2 x * Since g   x    3  x    x  4 6   x 2 x   3 x4  x6  g x  , x2 then g is an even function and its graph is symmetric about the y-axis. x  x4 (iii) h  x  . x 1 2   x    x  4  x  x4 * Since h   x      h x  ,   x 2  1 x 12 11  then h is neither even nor odd function and its graph has no symmetry. Increasing and Decreasing Functions : (i) A function f is called increasing on an interval I if f  x1   f  x 2  whenever x1  x 2 in I. (ii) A function f is called decreasing on an interval I if f  x1   f  x 2  whenever x1  x 2 in I. * The function f  x   x 2 is decreasing on    ,0  and increasing on 0 ,  . 12 Periodic Functions : A function f is called a periodic function with period T if f  x  nT   f  x  ; n  . Remark : * Trigonometric functions are periodic functions. 13 1.2 Some Essential Functions : (a) Linear Functions : A graph of a linear function is a straight line. The slope- intercept form of the equation of a straight line is y  f  x   mx  b , where m is the slope of the line and b is the y-intercept. * y  3x 2  m 3 , b  2.     ,   , Range       ,  . Domain   (b) Power Functions : A function of the form f  x   x n , where n is a constant , is called a power function. (i) n  1  Domain  , Range  . 14 (ii) n  2 , 4 , 6 ,...  Domain   , Range  0 , . (iii) n  3 , 5 , 7 ,...  Domain   , Range  . 15  1 1 1 (iv) n  , , ,...  Domain  0 ,  , Range  0 , . 2 4 6 1 1 1 (v) n  , , ,...  Domain   , Range  . 3 5 7 (vi) n   1 ,  3 ,  5 ,...  Domain   0  , Range   0 . 16 (vii) n   2 ,  4 ,  6 ,...  Domain   0  , Range  0 , . 3 2 (viii) n  ,  Range  0 , . 2 3 Domain  0 , . Domain  . (c) Polynomials : A function f is called a polynomial if f  x   an x n  an  1 x n 1 ...  a 2 x 2  a1 x  a0 , where n is a nonnegative integer and an , an  1 ,... , a0 are constants. Remarks : (1) f  x   2 x  1 , g  x   3 x 2  x  1 , h  x   5 x 3  x 2  2 x  7 are examples of polynomials. (2) The domain of a polynomial function is      , . 17 (d) Rational Functions : A rational function f is a ratio of two polynomials P x  f x   , Q x  where P and Q are polynomials. Remarks : 2 x x x2 5 (1) f  x   , g x   , h x   x 1 2 x 1 3 x4  x 3  x  1 are examples of rational functions. (2) The domain of a rational function is    zeros of Q . (e) Algebraic Functions : A function f is called an algebraic function if it can be constructed from polynomials using algebraic operations (addition, subtraction, multiplication, division, and taking roots). Remarks : x 1 1 (1) f  x   x 2  1 , g  x    x 3 , h x   3 2  x2  x x 1 are examples of algebraic functions. (2) A root function is in the form f  x   n g  x  , where g  x  is a polynomial function , n  . (3) The domain of root function is   if n is od d.   the solution set of ineq uality g  x   0 if n is even. 18 Example (1) : Find the domain of the following functions : x 5 (ii) g  x   x 2  x  2. (i) f  x  . x2  x  2 x 3 x 1 (iii) h  x  . (iv) u  x  . x2  x  2 x 2 Solution x 5 (i) f  x  .  Rational function x x 2 2 * To find zeros of denominator , put x2  x  2 0   x  1  x  2   0  x   1 or x  2. * D  f      1 , 2 . (ii) g  x   x 2  x  2.  Root function * g is defined for all real values of x such that : x2  x  2 0   x  1  x  2   0. (See Example (2) (i) – Precalculus Review). Sign of  x  1  Sign of  x  2  Sign of  x  1  x  2  D  g     ,  1   2 ,  *.     1 ,2  19  x 3 (iii) h  x  .  Algebraic function x x 2 2 * h is defined for all real values of x such that : x2  x  2 0   x  1  x  2   0. Sign of  x  1  Sign of  x  2  Sign of  x  1  x  2  D  h    ,  1   2 , *.     1 , 2  x 1 (iv) u  x  .  Algebraic function x 2 * u is defined for all real values of x such that : x 1 (1) x  2 0  (2) 0. and x 2 x 2. Sign of  x  1  Sign of  x  2   x 1  Sign of   x 2  20  D  u    ,  1   2 , *.     1 ,2  Example (2) : Find the domain of the following functions : x 1 (i) f  x  . (ii) g  x   1  3. 1 x 2 1 x 1 1 (iii) h  x  . (iv) u  x   x2 5 x 6. 2 1 x 3  1 x Solution x 1 (i) f  x  . 1 1 x 1 * f is defined for all real values of x such that : 1 (1) x 1 0  (2) 1 0 x 1 x   1. and 1 1 x 1 x 1   1 x  2. * D  f      2 ,  1 . 21  3 (ii) g  x   1 . x 2 * g is defined for all real values of x such that : 3 (1) x  2 0  (2) 1 0 x 2 x 2. and x 1 0. x 2 (See Example (1) (iv) – Section 1.2). Sign of  x  1  Sign of  x  2  x 1  Sign of   x 2  D  g    ,  1   2 , *.     1 ,2  1 (iii) h  x  . 2 3 1 x * h is defined for all real values of x such that : 2 (1) x 0.  (2) 3 1 0 and x 2 1  3 x 22  2 3 1  3 x 2  2  4 x 1 1 2. x 1  1 1 2 x and x 1 1 0  1  2 0 x x 1x 1 2x 0. 0. x x Sign of 1  x  Sign of 1  2 x  Sign of  x  Sign of  x  1x  Sign of  1 2x   x   Sign of    x  1  D  h    ,  1    ,   2  *.  1     1 ,   2 23  x2 5 x 6 (iv) u  x  . 1 x * u is defined for all real values of x such that : x2 5 x 6 (1) 1  x 0  (2) 0 and 1 x x 1.  x  2  x  3   0. 1 x Sign of  x  2  Sign of  x  3  Sign of 1  x Sign of  x  2  x  3  1 x * D  u      , 1   1 , 2   3 ,  . 24 Methods of Finding Range of Function : (1) Algebraic Method : By using simple logic obtain smallest value and largest value of the function. (2) Graphical Method : By sketching the graph of function (See Section 1.3 – Lecture Notes (2)) obtain the range of the function. (3) Derivative Method : By putting f '  x   0 obtain critical numbers , substitute in the function obtain local maximum and local minimum values and compare them with end points values to find range of function. Example (3) : Find the domain and the range of the following functions : (i) f  x   x 1. (ii) g  x   4  x. 1 3 (iii) h  x  . (iv) u  x  . 9 x 2 1 4 x 2 Solution (i) f  x   x 1. * f is defined for all real values of x such that : x 1 0  x 1  1x  x 1. or D  f      ,  1   1 ,   *.      1 ,1  * To find the range of f (Algebraic)  Smallest value At x   1  D  f   f  1   0. 25   Largest value As x    f x  . * R  f   0 ,  . (ii) g  x   4  x. * g is defined for all real values of x such that : (1) x 0.  (2) 4  x 0 and x 4 x  16. * D  g   0 ,16 . * To find the range of g (Algebraic)  Smallest value At x  16  D  g   g  16   0.  Largest value At x  0  D  g   g 0   2. * R  g   0 , 2 . 1 (iii) h  x  . 9 x 2 * h is defined for all real values of x such that : 9  x2 0  x2  9  x 3   3 x 3. * D  h    3 ,3 . 26 * To find the range of h (Algebraic)  Smallest value 1 At x  0  D  h   h 0  . 3  Largest value As x 3  or  3   D  h   h x  . 1  * R  g    ,. 3  3 (iv) u  x  . 1 4 x 2 * u is defined for all real values of x such that : (1) 4  x2 0  (2) 1  4  x 2  0 and x2   4 4  x2   1 x. x. * Du  . * To find the range of u (Algebraic)  Smallest value As x    u x  0.  Largest value At x  0  D u   u 0   1. * R  u   0 , 1 . 27 (f) Trigonometric Functions (i) Of an Acute Angle  : b c sin   csc   c b c cos   a sec   c a b a tan   cot   a b (ii) Of any Angle  : b r sin   csc   r b r cos   a sec   r a b a tan   cot   a b (iii) Of a Real Number x : The value of a trigonometric function at a real number x is its value at an angle of x radians. 1 1 sin  cos  (iv) csc   , sec   , tan   , cot  . sin  cos  cos  sin  28 Graphs of The Trigonometric Functions : Function Graph Domain and Range Domain       , . Range    1 ,1 . y  sin x Period  2 . Odd function ; sin   x    sin x. Domain       , . Range    1 ,1 . y  cos x Period  2 . Even function ; cos   x   cos x. Domain       n ; n   . y  tan x 2  sin x Range       , .  cos x Period  . Odd function ; tan   x    t an x. 29  Domain     n ; n   . y  cot x Range       , . cos x  sin x Period  . Odd function ; cot   x    cot x. Domain       n ; n   . y  sec x 2  1 Range     ,  1  1,    cos x      1 ,1 . Period  2 . Even function ; sec   x   sec x. Domain     n ; n   . y  csc x 1 Range     , 1  1,    sin x      1 ,1 . Period  2 . Odd function ; csc   x    csc x. 30 Identities of The Trigonometric Functions : (1) cos 2 x  sin 2 x  1 , 1  tan 2 x  sec 2 x , cot 2 x  1  csc 2 x. (2) Additional Formulas : sin  x  y   sin x cos y  cos x sin y , cos  x  y   cos x cos y  sin x sin y , tan x  tan y tan  x  y  . 1  tan x tan y (3) Double-Angle Formulas : sin 2 x  2 sin x cos x , cos 2 x  cos 2 x  sin 2 x  2 cos 2 x  1  1  2 sin 2 x , 2 tan x tan 2 x . 1  tan x 2 (4) Half-Angle Formulas : 1 sin 2 x  1  cos 2 x  , 2 1 cos 2 x   1  cos 2 x . 2 1 (5) sin 3 x   3 sin x  sin 3 x  , 4 1 cos 3 x   3 cos x  cos 3 x . 4 31  1 (6) sin x cos y  sin  x  y   sin  x  y   , 2  1 cos x cos y  cos  x  y   cos  x  y   , 2 1 sin x sin y  cos  x  y   cos  x  y  . 2 (g) Transcendental Functions : Transcendental functions include trigonometric , inverse trigonometric , exponential , and logarithmic functions , and many other functions as well. Functions that are not algebraic are called transcendental. Remark : ln x 1  log 5 x * f  x   sin x  , g x   5 x  3 x  2 , h x   x 1 tan x are examples of Transcendental functions. Example (4) : Study the symmetry and find the domain of the following functions : x2 1 x 1 (i) f  x  . (ii) g  x  . 1  2 cos x 1  t an x x (iii) h  x  . 1  csc x Solution 32  x2 1 (i) f  x  .  Transcendental function 1  2 cos x   x 1 2 x2 1 * Since f   x     f x  , 1  2 cos   x  1  2 cos x then f is an even function and its graph is symmetric about the y-axis. * f is defined for all real values of x such that : 1  2 cos x  0 1   cos x   x. 2 3  x   2 n ; n  . 3    *   D f       2 n  ; n   .  3  33  x 1 (ii) g  x  .  Transcendental function 1  t an x  x   1   x  1   g x * Since g   x    , 1  tan   x  1  tan x then g is neither even nor odd function and its graph has no symmetry. * g is defined for all real values of x such that :  (1) x   n ; n  .  (2) 1  tan x  0 2 and   tan x  1  x. 4  x  n ; n  . 4 34     * D  g       n ,  n ; n   . 2 4  x (iii) h  x  .  Transcendental function 1  csc x x x * Since h   x      h x  , 1  csc   x  1  csc  x  then h is neither even nor odd function and its graph has no symmetry. * h is defined for all real values of x such that : (1) x  n ; n  .  (2) 1  csc x  0 and   csc x   1  x  . 2  x   2 n ; n  . 2 35     *   D h     n  ,   2 n  ; n   .  2  36 1.3 New Functions from Old Functions (a) Transformations of Functions : (i) Vertical Shifting : Suppose c  0. To obtain the graph of y  f  x   c , shift the graph of y  f  x  a distance c units upward. y  f  x   c , shift the graph of y  f  x  a distance c units downward. (ii) Horizontal Shifting : Suppose c  0. To obtain the graph of y  f  x  c  , shift the graph of y  f  x  a distance c units to the right. y  f  x  c  , shift the graph of y  f  x  a distance c units to the left. 37 (iii) Vertical Scaling : Suppose c  1. To obtain the graph of y  c f x  , stretch the graph of y  f  x  vertically by a factor of c. 1 y  f x  , compress the graph of y  f  x  vertically by c a factor of c. 38 (iv) Horizontal Scaling : Suppose c  1. To obtain the graph of y  f c x  , compress the graph of y  f  x  horizontally by a factor of c. x y  f   , stretch the graph of y  f  x  horizontally by c  a factor of c. 39 (v) Reflecting : y   f  x  , reflect the graph of y  f  x  about the x-axis. y  f   x  , reflect the graph of y  f  x  about the y-axis. 40 Remarks : y1  x. y2  x 1. y3  x 1. y4  x. y5   x 1 y6   x 1   x  1 .   x  1 . * To sketch the graph of y  2 x  1. First Method : (1) y 1  x. (2) y 2  x 1. (3) y 3  2 x  1. 41 Second Method : 1 * y  2x 1  2 x. 2 1 1 (1) y 1  x. (2) y 2  x. (3) y 3  2 x. 2 2 (vi) Squaring Root : It is as if you take the square root of each value in f  x . Any positive value (part of curve above the x-axis) remains positive and will be equal to f  x . Any negative value (part of curve below the x-axis) will vanish. Points on the x-axis remain the same. y 1  f x . y2  f x . 42  (vii) Absoluting : (a) Taking the absolute value of the function f  x  : f  x  It is as you take the absolute value of each value in f  x . Any positive value or zero remains the same and any negative value becomes positive. (b) Taking the absolute value of the independent variable x : f  x  Point on the y-axis remains the same. The part of the curve at the left of the y-axis (x is negative) vanishes. The part of the curve at the right of the y-axis (x is positive) remains the same and reflects about the y-axis. y 1  f x . y 2  f x . y3  f  x . 43 (viii) Reciprocating : It is as if you take the reciprocal of each value in f  x . 1 y 1  f x . y2 . f x  General Guidelines of Reciprocating : (1) f  x  is positive 1 is positive. f x  reciprocating (graph above x-axis)  (graph above x-axis) (2) f  x  is negative 1 is negative. f x  reciprocating (graph below x-axis)  (graph below x-axis) (3) f is increasing 1 reciprocating is decreasing.  f 1 (4) f is decreasing reciprocating is increasing.  f 44  1 1 (5)  Vertical ,  Hole , 0 asymptote Hole 1  Hole on the x-axis Vertical asymp tote 1 1 (6)    ,    0 0 1 1 (7)  0  ,  0   Example (1) : Sketch the graphs of the following functions (in steps) and find their domains and ranges : (1) f 1  x   x 2. (2) f 2  x    x  1  2. (3) f 3  x   x 2  2 x  3. (4) f 4  x   1 2 2  x 2x 3.  1 (5) f 5  x   x 2  x  3. (6) f 6  x    x 2  2 x  3. 4 (7) f7  x   x 2  2 x  3. (8) f 8  x   x 2  2 x  3. (9) f 9  x   x 2  2 x  3. (10) f 10  x   x 2  2 x  3. 1 (11) f 11  x  . x 2x 3 2 Solution 45 (1) f 1  x   x 2. * D  f1   . * R  f 1   0 ,  . (2) f 2  x    x  1  2  f 1  x  1 . (Horizontal shifting f 1 to right) * D  f2   . * R  f 2   0 ,  . (3) f 3  x   x 2  2 x  3  Completing square  x  1  4 2  f2 x  4. (Vertical shifting f 2 downward) * D  f3   . * R  f 3    4 , . 46  1 (4) f 4  x   2  x2  2x 3  1  f3 x . 2 (Vertical compressing f 3 ) * D  f4   . * R  f4    2 , . 1 (5) f 5  x   x 2  x  3 4 2 x x   2 3 2  2  x  f3  . 2  (Horizontal stretching f 3 ) * D  f5   . * R  f 5    4 , . (6) f 6  x    x 2  2 x  3    x2  2x 3    f3 x . (Reflecting f 3 about the x-axis) * D  f6   . * R  f 6      ,4 . 47 (7) f7  x   x 2  2 x  3   x   2  x   3 2  f3  x . (Reflecting f 3 about the y-axis) * D  f7   . * R  f7     4 ,  . (8) f 8  x   x 2  2 x  3  f3 x . (Squaring root f 3 ) D  f 8      ,  1   3 ,   *.     1 ,3  * R  f 8   0 ,  . (9) f 9  x   x 2  2 x  3  f3 x . (Absoluting function f 3 ) * D  f9   . * R  f 9   0 ,  . 48 (10) f 10  x   x 2  2 x  3. 2  x 2 x 3  f3  x . (Absoluting independent variable x of f 3 ) * D  f 10   . * R  f 10     4 ,  . 1 (11) f 11  x  . x 2x 3 2 1 . f3 x  (Reciprocating f 3 ) * D  f 11       1 , 3 .  1  *  11  R f      ,0 .  4  49 Example (2) : Sketch the graphs of the following functions (in steps) , label each step with its equation , and find their domains and ranges : 1 1 (i) f  x   2 . (ii) g  x  . x 1 2  x 1 2 (iii) h  x   1 . 1x Solution 1 (i) f  x   2 . x 1 (1) y 1  x. (2) y 2  x  1. 1 1 (3) y 3 . (4) y 4  . x 1 x 1 50  1 (5) f  x   2 . x 1 * Df  . * R  f   1 , 2 . 1 (ii) g  x  . 2  x 1 (1) y 1  x. (2) y 2  x  1. (3) y 3   x  1. (4) y 4  2  x  1. 51  1 (5) g  x  . 2  x 1 * D  g      3 , 1 .  1 * R  g     0 , .  2 2 (iii) h  x   1 . 1x 1 1 (1) y 1 . (2) y 2 . x x 1 1 2 (3) y 3  . (4) y 4 . 1 x  x  1  1 x 52  2 (5) h  x   1 . 1x * D  h     1 . * R  h     1 . Example (3) : Sketch the graphs of the following functions (in steps) , label each step with its equation , and find their domains and ranges : 2 1 (i) f  x  . (ii) g  x   1 . 1 x 2x 3 2 (iii) h  x  . 1 1 x Solution 2 (i) f  x  . 1 x (1) y 1  x. (2) y 2   x. 53  1 (3) y 3  1  x. (4) y 4 . 1 x 2 (5) f  x  . 1 x * D  f   0 ,    1 . * R  f     0 , 2 . 1 (ii) g  x   1 . 2x 3 (1) y 1  x. (2) y 2  x 3. 54  1 (3) y 3  2 x  3. (4) y 4 . 2x 3 1 (5) g  x   1 . 2x 3  3  * D  g     ,.  2  * R  g   1 ,  . Another Method : 1 1/ 2 g x   1  1  (Try to solve). 2x 3 3 x 2 2 (iii) h  x  . 1 1 x 55  1 1 (1) y 1 . (2) y 2  . x x 1 1 (3) y 3  1 . (4) y 4  1 . x x 1 2 (5) y 5 . (6) h  x  . 1 1 1 1 x x * D  h      ,0   1 ,      0 , 1 . * R  h    0 , 2    2 ,    0 ,     2 . 56 Example (4) : Sketch the graphs of the following functions (in steps) , label each step with its equation , and find their domains and ranges : 1 (i) f  x  . 1 x2  2x (ii) g  x  . 1 4 x 2 Solution 1 1 (i) f  x     Completing square. x  2x 2 x  1  2 1 (1) y 1  x 2. (2) y 2   x  1  2. (3) y 3   x  1  2  1. (4) y 4   x  1   1. 2 1 (5) u  x  . x  1  2 1 * D  u       2 ,0 . * R  u   0 ,  . 57  1 (ii) g  x  . 1 4 x 2 (1) y 1  4  x 2. (2) y 2   4  x 2. 1 (3) y 3  1  4  x 2. (4) y 4 . 1  4 x 2 1 (5) g  x  . 1 4 x 2  * D  v    2 , 2    3 . * R  v   1 ,  . 58 Example (5) : Sketch the graphs of the following functions (in steps) , label each step with its equation , and find their domains and ranges : 1 1 (i) f  x  . (ii) g  x  . cos x  1 1  sin x 1 (iii) h  x  .   tan  x    2 Solution 1 (i) f  x  . cos x  1 (1) y 1  cos x. (2) y 2  cos x. 1 (3) y 3  cos x  1. (4) f  x  . cos x  1 * D  f     n ; n  . * R  f     , 1. 59  1 (ii) g  x  . 1  sin x (1) y 1  sin x. (2) y 2   sin x. 1 (3) y 3  1  sin x. (4) g  x  . 1  sin x   * D  g       2 n ; n   . 2  1  * R  g    ,. 2  60  1 (iii) h  x  .   tan  x    2   (1) y 1  tan x. (2) y 2  tan  x  . 2    (4) h  x   1 (3) y 3  tan  x  ..  2    tan  x    2  n  * D  h   ; n .  2  * R  h   0 ,  . 61 62