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