Functions and Their Graphs Lecture Notes PDF

Functions and their Graphs Definition 1 A function f : A B is a rule that assigns to each element x  A exactly one element y  B such that f (x )  y. x is called the independent variable (input) and y is called the dependent variable (output). Definition 2. Let f : A B be a function. Then, (i...

Functions and their Graphs Definition 1 A function f : A B is a rule that assigns to each element x  A exactly one element y  B such that f (x )  y. x is called the independent variable (input) and y is called the dependent variable (output). Definition 2. Let f : A B be a function. Then, (i) A is called the domain of f , denoted by D f. (ii) B is called the co-domain of f. (iii) The set f (x ): x  A  is called the range of f , denoted by R f. Example 1: Let A  a,b ,c  , B  1,2,3 and f (a)  1, f (b )  2, f (c )  2. Then, Df  A co-domain of f  B and R f  1,2. Remark 1: R f  co-domain of f. Definition 3: A function f : A B is called real if both A and B are or subset of it. Example 3: Find the domain and range of the following functions. (i) f (x )  16  x 2 (ii) f (x )  4  x Answer. Put: 16  x 2  0  x 2  16  x 2  16  x 2  16  x 4  4  x  4 1   D f  [4,4]  R  [0,4]. f (ii) f (x )  4  x Answer. Put: 4  x  0 and x  0  x  4 and x  0  x  16 and x  0  D f  [0,16]  R f  [0,2]. Graphs of Functions. Definition 5 Let f : A B be a function. Then, the set (x , f (x ): x  A  is called the graph of f. Definition 9 A function f is called even if f (x )  f (x ), for all x  Df. A function f is called odd if f (x )  f (x ), for all x  Df. Remarks. (i) If the symmetry about y  axis, then the function is even. 2 (ii) If the symmetry about the origin point, then the function is odd. Example 10. Which of the following functions are even, odd, or neither? , and then discuss the symmetry of their graphs (if it exists): (i) f (x )  x 4  1 (ii) f (x )  x 5  x Answer. Answer. f (  x )  (  x )5  (  x ) f (x )  (x ) 4  1  x 5  x  x  1  f (x )  f (x ) is even. 4  ( x 5  x )  The symetric about y - axis.  f (x )  f (x ) is odd  The symetric about the origin point. (iii) f (x )  2x  x 2 Answer. f (x )  2(x )  (x )2  2x  x 2  f (x )  (2x  x 2 )  f (x )  f (x ) is neither even nor odd  No symmetry. Definition 11. A polynomial function is a function on the form: f (x )  an x n  an 1x n 1 .......  a0 , where n is a non-negative integer and a0 , a1,....., an . For example, the function f (x )  x 3  3x  1 is a polynomial function but the function f (x )  x 2  3x 2  2 is not a polynomial function. The domain of these functions is. 3 Definition 12 A rational function is a function on the form: g (x ) f (x )  , h (x ) where g (x ) and h (x ) are polynomial functions. x 3  3x  1 For example, the function f (x )  is a rational function but the x 2 4 x 2 function f (x )  is not a rational function. x 1 The domain of these functions is  x:h (x )  0 Example 11: Find the domain of the following functions 1 Draft (i) f (x )  x 2 x x x 0 2 Answer.  x (x  1)  0 Df   0,1  x  0 or x  1. 5x  4 x 2  3x  2  0 (ii) g (x )  2 x  3x  2  (x  1)(x  2)  0 Answer.  x  1 or x  2. D g   1, 2 (iii) h (x )  x 2  2x Answer. D h  since h (x ) is a polynomial function Defiition 13 Functions that can be constructed from polynomials by applying finitely many algebraic operations (addition, subtraction, multiplication, division, and root extraction) are called algebraic functions. 4 Example 12: Classify each equation as a polynomial, rational, algebraic, or not an algebraic function. (i) y  x  2 (ii) y  3x 4  x 1 x 2 5 (iii) y  5x 3  cos 4x (iv) y  2x  7 2 (v) y  3x  4x 2 Answer. (i) Algebraic (ii) Polynomial (iii) Not algebraic (iv) Rational (v) Rational Composite Functions. Definition 14. If f and g are functions, then the composite function f g is defined by (f g )(x )  f (g (x )). Example 15. If f ( x)  x and g ( x)  2  x , then find each of the following functions (i) f g (ii) g f (iii) f f (iv) g g (i) (f g )(x )  f ( g (x ))  f ( 2x )  2x  4 2x (ii) ( g f )(x )  g (f (x ))  g( x )  2 x 5 (iii) (f f )(x )  f (f (x )) f ( x )  x 4x (iv) ( g g )(x )  g ( g (x ))  g( 2x )  2 2x Example 16. Sketch the graph of the following functions, and then discuss the following: (a) Domain and range (b) The intercept with axes (c) The symmetry (d) The monotony. (i) f (x )  x 2 1 (ii) f (x )  x 2  4 (iii) f (x )  x 1  2 (i) f (x )  x 2 1 Answer. (a) Df  and R f  [1, ). 𝑥2 (b) There is no x  intercept. y  intercept: (0,1) (c) The symmetry about y  axis. (d) The function is increasing on [0, ) (1) (d) The function is decreasing on (,0] 𝑥2 + 1 1 6  (2) (ii) f (x )  x 2  4 Answer. (a) D (f )  and R (f )  [0, ). 𝑥2 (b) x  intercept: (2,0),(2,0) y  intercept: (0, 4) (c) The symmetry about y  axis. (1) 𝑥 2 -4 (d) The function is increasing on [2,0] [2, ) and decreasing on −2 2 (, 2] [0,2] −4 (2) 4 𝑥2 − 4 −2 2 (3) (iii) f (x )  x 1  2 Answer. (a) Df  [1, ) and R f  [2, ). 𝑥 (b) There is no x  intercept 7  There is no y  intercept (1) (c) No symmetry 𝑥−1 1 (d) The function is increasing on [1, ) (2) 2 𝑥−1+2 1 (3) Inverse Functions To find the inverse of the function f (x )  y : (i) Solve the equation f (x )  y for x , in terms of y. (ii) Use the symbol f 1 ( y ) to name the resulting expression in y. (iii) Replace y by x to get the formula for f 1 (x ). Remark: 1 If f : A B has an inverse function f : B A , then D f 1  R f and R f 1  Df Example: Find the inverse of the following functions, and the domain and range of the inverse function. (i) f (x )  3x  5 (ii) f (x )  x 3  2 (iii) f (x )  x 2  3, x  0 Exponential and Logarithm Functions. 8 Exponential Functions. Definition 5 An exponential function is a function of the form f (x )  a x , where a  0 and a  1. For Example. f (x )  2x f (x )  10x f (x )  e x , where e  2.71828 is the natural exponential. The following is the graph of an exponential function: Laws of Exponents. If a, b  0 and a, b  1 , then: ax (i) a x.a y  a x  y ; (ii)  ax  y ; a y (iii) (a x ) y  a xy ; (iv) (ab )x  a x.b x. Logarithmic Functions. Definition 6 A logarithm function is a function of the form f (x )  loga x , where, a > 0 and a  1.. The following is the graph of a logarithm function: 9 Laws of Logarithms. If x , y  0 , then: (i) loga x  loga y  loga xy x (ii) loga x  loga y  loga y (iii) r loga x  loga x r where r . Natural Functions. The natural logarithm function is loga x  ln x Remark 2. ln (e x )  x , for all x  and e ln x  x , for all x  0. Example 6 Solve the following equations. (i) e 53x  10 (ii) 2ln x  1 (iii) ln(2x 1)  2  ln x Trigonometric Functions. f (x )  sin x ,cos x ,tan x ,cot x ,sec x ,csc x The following is the graph of trigonometric functions 10 Domain restrictions that make the trigonometric functions injective (one to one): 11  Inverse of Trigonometric Functions. f (x )  sin 1 x ,cos1 x , tan 1 x ,cot 1 x ,sec1 x ,csc1 x The following is the graph of inverse of trigonometric functions: Example 1: Rewrite the following expressions in terms of x. 1 x 1 (i) sin (tan x) (ii) tan (arc cos ) (iii) cos (2 tan x) 3 (i) sin (tan 1 x ) Answer. 12 Let y  tan 1 x  tan y  x x  sin y  x 2 1 x y 1 x (ii) tan (arc cos ) 3 Answer. x x Let y  cos 1  cos y  3 3 9x2 3  tan y  9 − 𝑥2 x y x (iii) cos(2tan 1 x ) Answer. Let y  tan 1 x  tan y  x  cos 2 y  cos 2 y  sin 2 y 1 x ( )2  ( )2 x 1 x 2 1 x 2 1 x2 y   0 1 x 2 1 x 2 1 1 x 2  1 x 2 13  Hyperbolic Functions 1 1 sinh x  (e x  e  x ), cosh x  (e x  e  x ) 2 2 x sinh x e  ex tanh x   cosh x e x  e  x 1 cosh x e x  e  x coth x    tanh x sinh x e x  e  x 1 2 sech x   x cosh x e  e  x 1 2 csch x   x sinh x e  e  x The following is the graph of hyperbolic functions: Example 2: Prove that: 14 (i) sinh (x )   sinh x (ii) cosh x  sinh x  e x (iii) sinh (x  y )  sinh x cosh y  cosh x sinh y (iv) sinh 2x  2sinh x cosh x Example 3: 3 Find the values of the other hyperbolic functions if sinh x . 4 Inverse Of Hyperbolic Functions sinh 1 x ,cosh 1 x , tanh 1 x ,coth 1 x ,sech 1 x ,csch 1 x 15 Example 4: Prove that: 1 1 1 1 2 (i) sech x  cosh ( ) (ii) sinh x  ln ( x  x  1), x  x 1 1 1 x (iii) tanh x  ln , x 1 2 1 x 16