Daisy Song and Emily Shifflett

5.1: Composite Functions 5.1: Composite Functions 5.2: One-to-One Functions; Inverse Functions 5.2: One-to-One Functions; Inverse Functions 5.3: Exponential Functions 5.3: Exponential Functions 5.4: Logarithmic Functions 5.4: Logarithmic Functions 5.5: Properties of Logarithms 5.5: Properties of Logarithms 5.6: Logarithmic and Exponential Equations 5.6: Logarithmic and Exponential Equations 5.7: Compound Interest 5.7: Compound Interest 5.8: Exponential Growth and Decay Models; Newton’s Law; Logistic Growth and Decay Models 5.8: Exponential Growth and Decay Models; Newton’s Law; Logistic Growth and Decay Models 5.9: Building Exponential, Logarithmic, and Logistic Models from Data 5.9: Building Exponential, Logarithmic, and Logistic Models from Data

(f °g)(x)= f(g(x)) (f °g)(x)= f(g(x)) Domain (f °g) is all numbers in the domain of g(x) that coincide with domain of f(x). Domain (f °g) is all numbers in the domain of g(x) that coincide with domain of f(x).

Function is one-to-one is x 1 and x 2 are two different inputs of a function and x 1 ≠ x 2 Function is one-to-one is x 1 and x 2 are two different inputs of a function and x 1 ≠ x 2 A function is one-to-one if, when graphed, a horizontal line ever only hits one point. A function is one-to-one if, when graphed, a horizontal line ever only hits one point. A function that is increasing on an interval I is a one-to-one function on I. A function that is increasing on an interval I is a one-to-one function on I. A function that is decreasing on an interval I is a one-to-one function on I. A function that is decreasing on an interval I is a one-to-one function on I.

Inverse function of f: correspondence of the range of f back to the domain of f Inverse function of f: correspondence of the range of f back to the domain of f Domain f= range f -1, range f= domain f -1 Domain f= range f -1, range f= domain f -1 x 1 =y 1, x 2 =y 2 y 1 =x 1, y 2 =x 2 where y 1 ≠y 2 and x 1 ≠x 2 x 1 =y 1, x 2 =y 2 y 1 =x 1, y 2 =x 2 where y 1 ≠y 2 and x 1 ≠x 2 Verifying the Inverse: f(f -1 (x))= x and f -1 (f(x))=x Verifying the Inverse: f(f -1 (x))= x and f -1 (f(x))=x Finding the Inverse Function: Finding the Inverse Function:x=Ay+BAy+B=xAy=x-B y= (x-B)/A

f(x)=a x where a>0 and a≠1 f(x)=a x where a>0 and a≠1 Laws of Exponents Laws of Exponents (a s )(a t )= a s+t (a s ) t =a st (ab) s =a s b s 1 s =1 a -s =1/a s = (1/a) s a 0 =1

For f(x)=a x, a>0 and a≠1, if x is any real number then f(x+1)/f(x)=a For f(x)=a x, a>0 and a≠1, if x is any real number then f(x+1)/f(x)=a f(x)=a x +n f(x)=a x +n n moves the function up and down the y-axis n moves the function up and down the y-axis The number e: (1+(1/n)) n =e The number e: (1+(1/n)) n =e y=e x-n y=e x-n n stretches the curve n stretches the curve If a b =a c then b=c If a b =a c then b=c

y=log a x if and only if x=a y y=log a x if and only if x=a y Domain: 0<x<∞ Range: -∞<x<∞ Domain: 0<x<∞ Range: -∞<x<∞ y=lnx if and only if x=e y y=lnx if and only if x=e y Graphing Logarithmic Functions: Graphing Logarithmic Functions: Plot y=a x of y=log a x function Plot y=a x of y=log a x function Draw the line y=x Draw the line y=x Reflect graph Reflect graph

Properties of Logarithmic Functions: Properties of Logarithmic Functions: Domain is x>0, Range is all real numbers Domain is x>0, Range is all real numbers x-intercept is 1. There is no y-intercept x-intercept is 1. There is no y-intercept The y-axis is a vertical asymptote The y-axis is a vertical asymptote The function is decreasing if 0 1 The function is decreasing if 0 1 The graph contains the points (1,0), (a,1), and (1/a, 1) The graph contains the points (1,0), (a,1), and (1/a, 1) The graph is smooth and continuous, with no corners or gaps The graph is smooth and continuous, with no corners or gaps

Graphing Inverse Logarithmic Functions: Graphing Inverse Logarithmic Functions: Find domain of f(x) and graph Find domain of f(x) and graph Find the range and vertical asymptote Find the range and vertical asymptote Find f-1(x) Find f-1(x) Use f-1 to find the range of f Use f-1 to find the range of f Graph f-1 Graph f-1

log a 1=0 log a 1=0 log a a=1 log a a=1 a log a M =M a log a M =M log a a r =r log a a r =r log a (MN)= log a M+log a N log a (MN)= log a M+log a N log a (M/N)= log a M-log a N log a (M/N)= log a M-log a N log a M r = rlog a M log a M r = rlog a M log a M= log b M/log b a (a≠1, b≠1) log a M= log b M/log b a (a≠1, b≠1)

y=log a x = x=a y if a>0, a≠1 y=log a x = x=a y if a>0, a≠1 If log a M=log a N then M=N if M,N, and a are positive and a≠1 If log a M=log a N then M=N if M,N, and a are positive and a≠1 If a u =a v then u=v if a>0, a≠1 If a u =a v then u=v if a>0, a≠1 Graphing calculator can be used to graph both sides of the equation and find the intersect Graphing calculator can be used to graph both sides of the equation and find the intersect

Simple Interest Formula: I=Prt Simple Interest Formula: I=Prt Interest=(amount borrowed)(interest rate)(years) Interest=(amount borrowed)(interest rate)(years) Payment periods Payment periods Annually: once per year Annually: once per year Semiannually: twice per year Semiannually: twice per year Quarterly: four times per year Quarterly: four times per year Monthly: 12 times per year Monthly: 12 times per year Daily: 365 times per year Daily: 365 times per year

Compound Interest: I new =P old+I old Compound Interest: I new =P old+I old Compound Interest Forumula: A=P(I+(r/n)) nt Compound Interest Forumula: A=P(I+(r/n)) nt Amount= (Principal amount)(Interest+(rate/times per year)) (number per year)(years) Amount= (Principal amount)(Interest+(rate/times per year)) (number per year)(years) Effective interest: annual simple rate of interest= compounding after one year Effective interest: annual simple rate of interest= compounding after one year Present value: time value of money Present value: time value of money P=A(1+(r/n) -nt if compound P=A(1+(r/n) -nt if compound P=Ae -rt if continuously compounded P=Ae -rt if continuously compounded

Law of Uninhibited Growth: A(t)=A 0 e kt A 0 =amount when t=0 k≠0 Law of Uninhibited Growth: A(t)=A 0 e kt A 0 =amount when t=0 k≠0 Amount(time)= (original amount)e (constant)(time) Amount(time)= (original amount)e (constant)(time) Uninhibited Radioactive Decay, and Uninhibited Growth of Cells follow this model Uninhibited Radioactive Decay, and Uninhibited Growth of Cells follow this model Half-life: time required for ½ of the substance to decay Half-life: time required for ½ of the substance to decay Newton’s Law of Cooling: u(t)= T+(u 0 -T)e kt Newton’s Law of Cooling: u(t)= T+(u 0 -T)e kt Temperature(time)=constant temp+ (original temp- constant temp)e (constant)(time) Temperature(time)=constant temp+ (original temp- constant temp)e (constant)(time)

Logistic Model: P(t)=c/1+ae -bt Logistic Model: P(t)=c/1+ae -bt Population(time)=carrying capacity/1+(constant)e -constant(time) Population(time)=carrying capacity/1+(constant)e -constant(time) c=carrying capacity P(t) approaches c as t approaches ∞ c=carrying capacity P(t) approaches c as t approaches ∞ Describes situations where growth and decay is limited Describes situations where growth and decay is limited b>0 growth rate, b 0 growth rate, b<0 decay rate

Properties of Logistic Growth Function: Properties of Logistic Growth Function: Domain is all real numbers Domain is all real numbers Range 0<y<c Range 0<y<c No x-intercepts, y-intercept at P(0) No x-intercepts, y-intercept at P(0) Two horizontal asymptotes at y=o and y=c Two horizontal asymptotes at y=o and y=c P(t) is increasing if b>0 P(t) is increasing if b>0 P(t) is decreasing if b<0 P(t) is decreasing if b<0 Inflection point= point where the graph changes direction (up to down for growth models and down to up for decay models)= P(t) ½ c Inflection point= point where the graph changes direction (up to down for growth models and down to up for decay models)= P(t) ½ c Smooth and continuous graph, no corners or gaps Smooth and continuous graph, no corners or gaps

Same as finding line of best fit Same as finding line of best fit Draw a scatter diagram of the data to see the general shape to determine what type of function to use Draw a scatter diagram of the data to see the general shape to determine what type of function to use Many calculators have a Regression option to fit an equation to data Many calculators have a Regression option to fit an equation to data To fit: enter data into calculator. Use correct regression to calculate model. To fit: enter data into calculator. Use correct regression to calculate model.