Lecture Notes on Function Types PDF

Function Types Table of Contents Function Types................................................................................................................................ 1 Exponential and Logarithm Functions......................................................................................

Function Types Table of Contents Function Types................................................................................................................................ 1 Exponential and Logarithm Functions....................................................................................... 1 Exponential Functions............................................................................................................ 1 Logarithmic Functions............................................................................................................ 2 Natural Functions....................................................................................................................... 3 Trigonometric Functions............................................................................................................. 5 Inverse of Trigonometric Functions........................................................................................... 6 Hyperbolic Functions.................................................................................................................. 8 Inverse of Hyperbolic Functions............................................................................................... 11 Exponential and Logarithm Functions. Exponential and logarithmic functions are powerful tools for modeling real-world situations. They help us understand growth, decay, and complex relationships in various fields like biology, finance, and physics. These functions allow us to predict future values and analyze trends in data. Applications of these functions bring math to life, showing how abstract concepts apply to everyday scenarios. From calculating compound interest to measuring earthquake intensity, exponential and logarithmic models provide insights into diverse phenomena. Understanding these applications enhances our ability to interpret and solve real-world problems. 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, 1 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 ; ay (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: 2 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 . Find the value of log5 (1/25). Given: log5 (1/25) By using the property, Logb (m/n)= logb m – logb n log5 (1/25) = log5 1 – log5 25 log5 (1/25) = 0 – log5 52 log5 (1/25) = -2log55 log5 (1/25) = -2 (1) [By using the property loga a = 1) log5 (1/25) = -2. Solve for x in log2 x = 5 his logarithmic function can be written In the exponential form as 25 = x Therefore, 25= 2 × 2 × 2 × 2 × 2 = 32, x= 32. Natural Functions. The biological sciences the mechanical revolution had a less dramatic impact on the use of natural function concepts in scientific explanation. The natural logarithm function is loga x  ln x 3  ln (mn) = ln m + ln n  ln (m/n) = ln m - ln n  ln mn = n ln m  ln a = (log a) / (log e)  ln e = 1  ln 1 = 0 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 Answer. 4 (i) e 53x  10 (ii) 2ln x  1 e 53x  10 ln x  1 2 5  3x  ln10 1  3x  5  ln10  x e 2 5 ln10 x   3 3 Trigonometric Functions. f (x )  sin x ,cos x , tan x ,cot x ,sec x ,csc x The following is the graph of trigonometric functions Domain restrictions that make the trigonometric functions injective (one to one): 5 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: 6 Example 7: Rewrite the following expressions in terms of x. 1 x (i) sin (tan x) (ii) tan (arc cos ) 3 Answer. (i) sin (tan 1 x ) 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 − 𝑥 2 x y x 7 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: 8 Example 8: Prove that: (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 4- Prove that: (i) sinh (x )   sinh x Proof. 1 1 1 sinh x  (e x  e  x )  sinh (x )  (e  x  e x )   (e x  e  x )   sinh x 2 2 2 (ii) cosh x  sinh x  e x Proof. 1 1 cosh x  (e x  e  x ), sinh x  (e x  e  x ) 2 2 1 1  cosh x  sinh x  (e x  e  x )  (e x  e  x ) 2 2 1 1 1 1  e x  e x  e x  e x  e x. 2 2 2 2 (iv) sinh (x  y )  sinh x cosh y  cosh x sinh y Proof. 1 1 cosh x  (e x  e  x ), sinh x  (e x  e  x ) 2 2 1 L.H.S  sinh (x  y )  (e x  y  e  ( x  y ) ) 2 9 R.H.S  sinh x cosh y  cosh x sinh y 1 1 1 1  (e x  e  x ). (e y  e  y )  (e x  e  x ). (e y  e  y ) 2 2 2 2 1 1  (e x  e  x )(e y  e  y )  (e x  e  x )(e y  e  y ) 4 4 1  e x  y  e x  y  e  x  y  e  x  y  e x  y  e x  y  e  x  y  e  x  y  4 1   2e x  y  2e  ( x  y )  4 1  (e x  y  e  x  y )  L.H.S 2 (iv) sinh 2x  2sinh x cosh x Proof. sinh (x  y )  sinh x cosh y  cosh x sinh y Put y  x  sinh (x  x )  sinh x cosh x  cosh x sinh x  sinh 2x  2sinh x cosh x. Example 9: 3 Find the values of the other hyperbolic functions if sinh x . 4 Answer. cosh 2 x  sinh 2 x  1 1 1 sinh x 1 csch x  sech x  tanh x  coth x  cosh x  sinh x  1 2 2 sinh x cosh x cosh x tanh x 4 4 3 5 cosh x  1  sinh 2 x  csch x   sech x   tanh x   coth x  3 5 5 3 3  1  ( )2 4 5  cosh x  4 10 Inverse of Hyperbolic Functions sinh 1 x ,cosh 1 x , tanh 1 x ,coth 1 x ,sech 1 x ,csch 1 x Important Rules 11 (1) sin 2 x  cos 2 x  1 (2) 1  tan 2 x  sec 2 x (3) 1  cot 2 x  csc 2 x (4) sin ( x  y )  sin x cos y  cos x sin y (5) cos ( x  y )  cos x cos y sin x sin y tan x  tan y (6) tan ( x  y )  1 tan x tan y (7) sin 2x  2sin x cos x (8) cos 2x  cos 2 x  sin 2 x  2 cos 2 x  1  1  2sin 2 x 2 tan x (9) tan 2x  1  tan 2 x (10) cosh 2 x  sinh 2 x  1 (11) 1  tanh 2 x  sech 2 x (12) coth 2 x  1  csch 2 x (13) sinh ( x  y )  sinh x cosh y  cosh x sinh y (14) cosh (x  y )  cosh x cosh y  sinh x sinh y tanh x  tanh y (15) tanh ( x  y )  1  tanh x tanh y (16) sinh 2x  2sinh x cosh x (17) cosh 2x  cosh 2 x  sinh 2 x  2sinh 2 x  1  2 cosh 2 x  1 2 tanh x (18) tanh 2x  1  tanh 2 x 12  Opposite tan x  Adjacent Opposite sin x  Hypotenuse Opposite Adjacent cos x  Hypotenuse x Adjacent 13