Exponential and Logarithmic Functions Chapter 4.1 PDF

Chapter4: Exponential and Logarithmic Functions 4.1 : Exponential Functions Definition: The exponential function 𝑓 with base 𝒃 is defined by 𝒇(𝒙) = 𝒃𝒙 𝒐𝒓 𝒚 = 𝒃𝒙 where b is a positive constant other than 1. (𝒃 > 0 , 𝑏 ≠ 1)....

Chapter4: Exponential and Logarithmic Functions 4.1 : Exponential Functions Definition: The exponential function 𝑓 with base 𝒃 is defined by 𝒇(𝒙) = 𝒃𝒙 𝒐𝒓 𝒚 = 𝒃𝒙 where b is a positive constant other than 1. (𝒃 > 0 , 𝑏 ≠ 1). ftp.mj 𝑥 is any real number. z.gg yy Domain of exponential function 𝒇(𝒙) = 𝒃𝒙 : all real numbers (R). tea Range of exponential function 𝒇(𝒙) = 𝒃𝒙 : (0, ∞) Examples : 1 - 𝑓(𝑥) = 2 , 𝑔(𝑥) = 10 , ℎ(𝑥) = 𝜋 , 𝑗(𝑥) = , 𝑘(𝑥) = 3 standard form 2 transformed form- The function 𝒇(𝒙) = 𝒆𝒙 is called a natural exponential function. The irrational number 𝑒 ≈ 2.72 is called a natural base. so Examples of non exponential functions: X isis isis seem is 𝑔(𝑥) = (−1) , 𝑓(𝑥) = 𝑥 , 𝑘(𝑥) = 1 , 𝑔(𝑥) = (−4) , H(x) = 𝑥 I a ↳ # ↳ & ↓ number Variable number negative base I negative variable =  Evaluating an exponential function: 𝐿𝑒𝑡 𝑔(𝑥) = (1.56) evaluate 𝑔(4) = (1 56)" 5 922. =. go.xss.sn Example 1: Approximate each number using a calculator. Round your answer to three decimal places. 0 125 5) 4 =.. 38690 387 9) 𝑒 = 0.. 1 Chapter4: Exponential and Logarithmic Functions 4.1 : Exponential Functions 10.67  Graphing Exponential Functions: Example 2: increasing 13) Graph 𝑓(𝑥) =. Then find domain, range and the equation of asymptote. stander form 𝑥 𝑓(𝑥) (𝑥, 𝑦) - O 1(z)* ()" = (z) (z) = = = 0 7 1. (1 : 10 , 11 0. 7) - * ⑨ ⑧ 1 (z) (2) = = 1 5. (1 1. 5) , because b > · The graph is increasing - ·& Asymptote 𝒇(𝒙) Domain - d) , or IR Range 10 d) , Horizontal on the X-axis Asymptote 0 y = H.A b 7 I 0 > b > Increasing Decreasing - - 2 Chapter4: Exponential and Logarithmic Functions 4.1 : Exponential Functions anreasing o.sc 17) Graph 𝑓(𝑥) = ( 0.6). Then find domain, range and the equation of asymptote 𝑥 𝑓(𝑥) (x, y) · :1 - I 10. 6)" = (0 67.. 66 (-1 1. 66) , 10 6 : 10 % 1 (0 1) = 0 =.. * , ⑧ 10 6)" 10 6 & I. =. = 0 6. (1 , 0. 6) - * because O >b > · The graph is decreasing 𝒇(𝒙) Domain 1-0 , 0 or IR Range 10 %) , Horizontal on the X-axis Asymptote 0 y = H.A 3  Chapter4: Exponential and Logarithmic Functions 4.1 : Exponential Functions s o w team sumn's H Transformation of Exponential Function (𝑓(𝑥) = 𝑏 ) Transformation Equation Description Vertical Shift 𝑔(𝑥) = 𝑏 + 𝑐 * up (𝑥, 𝑦) → (𝑥, 𝑦 + 𝑐) E shift in y 𝑔(𝑥) = 𝑏 − 𝑐 ↳ down (𝑥, 𝑦) → (𝑥, 𝑦 − 𝑐) Horizontal Shift 𝑔(𝑥) = 𝑏 * Teff (𝑥, 𝑦) → (𝑥 − 𝑐, 𝑦) shift in X 𝑔(𝑥) = 𝑏 Fight - (𝑥, 𝑦) → (𝑥 + 𝑐, 𝑦) Reflection about x-axis ( y) - 𝑔(𝑥) = −𝑏 (𝑥, 𝑦) → (𝑥, −𝑦) 𝑔(𝑥) = 𝑏 (𝑥, 𝑦) → (−𝑥, 𝑦) Reflection about y-axis ( x) - Vertical stretching or shrinking 𝑔(𝑥) = 𝑐𝑏 (𝑥, 𝑦) → (𝑥, 𝑐𝑦) Horizontal stretching or 𝑔(𝑥) = 𝑏 8 𝑥 (𝑥, 𝑦) → ( , 𝑦) shrinking 𝑐8 4  Transformations Shifts -- Hovizauntly vertically shift in X shift in y --- -- t - down = t Up Left right Reflection --- Reflection about X-axis Reflection about y-axis - Y g(x) f(x) f)0 x) g(x) = - = - - X f(x) g(x) f(x) 9)(x) (x , y) - (X , y) - (x , y) - ) - X , y) VerticallyStretching Shrinking Graphs --- number ("C : any 0xC > 0 < Multipling y by f(x) g(x) Stretch (x y) - (X c f) Shrink ,. , Chapter4: Exponential and Logarithmic Functions 4.1 : Exponential Functions Example3: 201intrease 29) Begin by Graphing 𝑓(𝑥) = 2.then use transformation of this graph to graph given function. Give the equation of the asymptotes. Use the graph to determine each function’s domain and range - - a) ℎ(𝑥) = 2 −1 2x +2 𝑥 𝑓(𝑥) = 2 (𝑥, 𝑦) h(x) = - 1 (x - 2 , y - 1) & ⑨ & & & ↑ - 2 ( 1) 1 t z( z) 2 - - - - , * ( 3 E) = , - , - - 20 , 2 1 1) 20 , 1 - 10 - 8 = I (-2 0) , 2 (1 2 2 1) 2 (1 2) - - I , = , ( - 1 , 1) 𝑓(𝑥) = 2 ℎ(𝑥) = 2 −1 Domain ( - 0 , 6) Domain ( - w , d Range 10 0) Range C- 1 G , , Horizontal Horizontal Asymptote 0 y - = Asymptote = H.A y H.A 5 Chapter4: Exponential and Logarithmic Functions 4.1 : Exponential Functions Example 4: Begin by Graphing 𝑓(𝑥) = 𝑒.then use transformation of this graph to graph given function. Give the equation of the asymptotes. Use the graph to determine each function’s domain and range %.. 𝑔(𝑥) = 𝑒 +2 ⑨ X - e f(x) = 0 =. eX(X 4 ( , 0 , y)(. 4) - (1 X , , 2 y + z). 4) - & ⑨ -D · · & 8 e = 1 10 , 1) (0 , 3) I e = 2. 4 (1 , 2. 4) (1 , 4. 7) 𝑔(𝑥) = 𝑒 +2 Domain ( - 0 , 0) Range (2 , 8) Horizontal Asymptote H.A 2 y = 6 Chapter4: Exponential and Logarithmic Functions 4.1 : Exponential Functions Example5: Begin by Graphing 𝑓(𝑥) =.then use transformation of this graph to graph given function. Give the equation of the asymptotes. Use the graph to determine each function’s domain and range 1 𝑔(𝑥) = −3 +1 2 & 𝑥 𝑓(𝑥) (𝑥, 𝑦) (x+ 1 , - 3y + 1) ()" ⑧ (+1 3(2) 1) (-1 2) - + 2 , - = 5) - (0 , ⑨ · - , 1) (2) 10 + 1 , 1) 3(1) + X-- D 0 1 (0 - = 2) ⑨ (1 , - (1 + 1 - 3(z) 1) (2) = + 1 11 , 2) (2 , - 0. 5) & & 𝑓(𝑥) 𝑔(𝑥) Domain ( - 0. 6) Domain ( - 0 , 6) Range 10 , %) Range ( - 0 , 1) Horizontal Horizontal Asymptote 8 Asymptote - 3(0) +1 = 1 y = H.A H.A y = 1 7 Chapter4: Exponential and Logarithmic Functions 4.1 : Exponential Functions Example 6: 61) Give the equation of the exponential function whose graph is shown. bor by form ((X) y = standard = ⑧ the · f(x) = " b = 4 = b - 4 = b using Fre point (1 , 4) * The final answer is f(x) = y · Example: ⑧ 8 Chapter4: Exponential and Logarithmic Functions 4.1 : Exponential Functions Example & Extra find domain and : 3 , Range asymptote X+ * 1 · g(x) e + g(x) 1(2) = + 3 - = · 2 Domain : )-O , 6) or Domain : 1-0 , 8) or R Range (1 Range (3 8) : 4) : , , = 3 Asymptote y Asymptote y : : = f(x) 574(1 =. 026)" = 574 f(x) = 574(1 026)*. - 1 , 148 with decimals 2001 - 1954 +X - = 27 8(x) 574(1 026)54 =. - 2295 2028 1974 - - X = 54 574(1 026)" R f(x) =. - + 4590 2055 - 1974 - X = 8) · Doubles up (get multiplied by 2) · Increases by the double 9 Chapter4: Exponential and Logarithmic Functions 4.1 : Exponential Functions 0 514 f(x) 80e - 80(1) 100. = + 20 -+ + 20 - % X = 0 0 51" 1 f(x) 69 -. 809 + 20 % - X = = 0 514) f(x) 31 % - Y 802 + 20 -. X = = 0 5152 52 f(x) -. X = = 80e + 20 - 20 % 10