
Chapter 1: Section 1.6 Inverse Functions and Logarithms Definition: Afunctionf(x)iscalled if it never takes on the same value twice; that is f(x ) = f(x )wheneverx = x . 1 6 2 1 6 2 Example 1: Horizontal Line Test: Afunctionisone-to-oneifandonlyifnohorizontallineintersectsitsgraph more than once. Example 2: Which of the following functions are one-to-one? y y y = x3 1 y = x4 x 0 1 1 x 0 1 Definition: If a function f is a one-to-one function with domain A and range B,thenits , 1 denoted as f − ,hasadomainB and range A and is defined as 1 f(x)=y x = f − (y) () 1 for any y in B.Thus,if(a, b)isapointonthegraphoff,then(b, a)isapointonthegraphoff − . 1 Thus, if f is one-to-one, then the graph of its inverse, f − ,isthereflectionabouttheliney = x. Chapter 1: Sec1.6, Inverse Functions and Logarithms Example 3: If f is a one-to-one function and given f(1) = 3 and f(3) = 1, then find 1 (a) f − (3) 1 (b) f − (f(3)) How to Find the Inverse Function of a One-to-One Function, y = f(x): (1) In the given function y = f(x), interchange x and y. (2) Solve the resulting equation for y to get the inverse function of f(x). 1 1 1 1 CAUTION: f − (x)isnotthesameas ,i.e.f − (x) = f(x) 6 f(x) 2x Example 4: Find the inverse of the function f(x)= and also state the domain and range of x +1 1 the function f and its inverse f − . Definition: The inverse of an exponential function, y = bx,iscalleda , provided b>0andb =1. 6 Properties of Logarithmic Function: If b, M,andN are positive real numbers, b =1,andx is a 6 real number. log 1=0 • b log b =1 • b log MN =log M +log N • b b b M log =log M log N • b N b − b ✓ ◆ log M N = N log M • b b log M =log N if and only if M = N • b b 2Spring2017,c Maya Johnson Chapter 1: Sec1.6, Inverse Functions and Logarithms Example 5: Express the given quantity as a single logarithm: 1 log (1 + x5)+ log (x) log (cos x) 10 2 10 − 10 Example 6: Find the exact value of the expression: log 12 log 28 + log 63. 3 − 3 3 Example 7: Solve the following for x: (a) log (x +2)+log (x 2) = log 12 10 10 − 10 2 (b) log x = log 27 2log 2 log 3 10 3 10 − 10 − 10 ,d27MHos,d25 - log ,o3 los ,o×= los - 19,03 losiox - 1%9-1%4 10518=1510/34 ) ⇒ los ,o×=1%Hq) -1%3 ⇒ 1%×=6%{ k¥3 : log ,o⇐, ×=# Definition: The logarithm with base e is called and has a special notation: loge x =lnx 3Spring2017,c Maya Johnson Chapter 1: Sec1.6, Inverse Functions and Logarithms Useful Logarithmic Property: For b and M positive real numbers, with b =1,blogb(M) = M 6 Example 8: Using the natural logarithm, find the value of x in the following: (a) ln(1 2x)=5 − (b) e2x+3 5=0 − Example 9: Consider the function f(x)=p4 e2x − (a) Find the domain of f 1 (b) Find f − - - ×z ↳ ⇒ ⇒ .e ×2=4-×2⇒ ×=f# ⇒ 4- x2 ) y.jo#inttahaeyxijI+eu=yy=hl4IZy ftp.hhj#1ne4=h( ×2=4 - e ⇒ - teas ted 2 ⇒ ×2+eY=4 zy=hy¥)z 1 (c) Find the domain of f − Need 4- x2>o : ) >O ( 2 - xX2tx - 3 X=2÷- ¥ }2012 Domain:f2€ 4Spring2017,c Maya Johnson Chapter 1: Sec1.6, Inverse Functions and Logarithms Change of Base formula: For any positive number a(a =1),thenwehave 6 ln x log x = a ln a Example 10: Given logb 3=0.6826 and logb 4=0.8614, where b is any positive integer, find the value of (a) logb 48 16 (b) log b b2 6 - bb2 = los los bhp loss =Tosy#b8 ;Ij2I*Io#ties÷ 5Spring2017,c Maya Johnson .
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