One-to-One Functions YouTube Video

Transcription

1 Section 9. One-to-One Functions YouTue Video A function in which each element in the range corresponds to one and only one element in the domain. Determine if the following are One-to-one functions:,, 4,, 5,,0,,,,,0,,,, If f(a) = f() implies a =, then f is one-to-one Determine if the following are One-to-one functions: y y

2 Inverse Functions YouTue Video If f is one-to-one, then an Inverse function, f (), eists Find the inverse y switching the and the y of the following:,, 4,, 5,4,0,,,, Find the inverse of the following: E/ y E/ y Try/ y 4 5 Try/ y, D: R: y 0 Finding the inverse of a graph: f ( ) f() = +

3 Graphing Eponential functions YouTue Video f f Section 9. f f f. Find the HA (horizontal asymptote) lael it y=. Translate or plot points like a mad man/woman. Graph the following, then find the domain, and range: HORIZONTAL ASYMPTOTE f ( ) f ( ) HA: y= HA: y= y y Domain: Range: Domain: Range:

4 Solving Eponential Equations y Changing Bases YouTue Video If 5, then =5 E/ 4 6 E/ 4 E/ 9 E/ 4 4 E/ 9 7 Try/ 7 Here is a special one: 8 4 4

5 Compounded Interest YouTue Video A= P= r= n= A P r n rt A Pe nt t=. How much will an account with $5000 compounded monthly with a rate of 5% have after 5 years?. How much will an account with $4000 compounded quarterly with a rate of 8.5% have after years?. How much will an account with $40 compounded semiannually with a rate of 8% have in 05? 4. How much will an account with $4000 compounded continuously with a rate of 8.5% have after years? 5. How much will an account with $,000 compounded daily with a rate of % have after 050? Another application: The atmospheric pressure P, in pounds per square inch, decreases eponentially with altitude h, in miles aove sea level, as given y P = 4.7 e 0.h. a) What is the atmospheric pressure at sea level? ) What is the atmospheric pressure at 4 miles? c) Is the atmospheric pressure increasing or decreasing as the height increases? 5

6 YouTue Video Section 9. Formulas Logartithms. y y. y y. y y r 4. c r c 5., ln e , 8. or 0 error, >0 a ln a 9. a ln Equation type Use formula Solve for or 4-8 gives an error when checked, so 4 is the only answer. X=4 Equation type Comine s This means the insides are = - gives an error message when checked, so No Solution Equation type 5 Need to ring the variale down using formula m 4 m. m ln6 ln 4 ( m )ln6 mln4 ( m ) m m m m.57 m e.006 q 0 m.006 lne ln0 Using formula ln0.006 ln ase eponent = result ase (result) = eponent Rewrite as a arithm Rewrite as an eponential equation

7 Simplify: = ln = Simplify: 00 0,000 5 ln e ln e 7 Change of Base: a ln a a ln Using your calculator: ln ln( ) ( 0) ln() ( ) 7

8 Solving: = ln = Solving part II: Special Properties 5.,, e ln or0 error, >0 4 lne

9 Review: Logarithms - Domain YouTue Video ln( ) ( 0) Find the domain of (+) Find the domain of (-) Find the domain of (-4) Find the domain of () A study showed that the numer of mice in and old aandoned house was approimated y the function M(t) = 6 4 (t + 4) Where t is the numer in months and t=0 corresponds to January 06. a) Find the domain and interpret what this means. ) How many mice are in the house in January 06? What does this mean? c) How many mice are in the house in June 06? d) How many mice might there e in the house in January 07? 9

10 Graphing Logarithmic functions YouTue Video What is the Domain of each:. Find the domain of the function.. Find the VA (Vertical asymptote) lael it =. Translate or plot points like a mad man/woman. Graph the following, then find the and y-int, domain, and range: y = (+) y = - 5 (-) VA: = VA: = Domain: Range: Domain: Range: 0

11 Section 9.4 Logarithms - Sum and Difference Formulas YouTue Video r y y y c r c y Write as the sum and difference of arithms y 5 y 5 y y r 5z Write as a single arithm 5 y Use 0 = 0.00 and 0 9 = to evaluate the following

12 Formulas involving common and natural arithms. Section 9.5 ph Scale: Pure water Acidic 7 Basic 0 4. PH H a) Find the PH for a liquid with a hydrogen ion concentration of. 8 X 0 moles per liter. Is it acidic(less than 7) or asic(greater than 7) ) Find the hydrogen ion concentration of a liquid with a PH of 8...Annual passenger traffic (in millions) at LAX can e approimated y the formula N( t) 58.7.ln(t ), where t is the numer of years since 990. a) What is the likely annual passenger traffic in 007? ) Find N (5); verally interpret. c) In what year will the annual passenger traffic reach 75 million?

13 Solving Logarithmic Equations YouTue Video Section 9.6 5

14 a) 5 5 ) ln e c) ( 4) 4 d) 4 e) ( 7) ( 9) f) ln5 4 ln ln 0 g) ( ) h) ( 7) ( 7) i) 4 hint: factor 4

15 Eponential equations - Solving using Logarithms YouTue Video e = = 6 + = = = 5 ) Try: a) ) c) 4 e d) 8 5e 5

16 Applications of Logarithms YouTue Video. Caron 4 dating. A monkey from a refuse deposit near the Strait of Magellan had 40 grams of the caron4 when it died. If the formula for amount for caron 4 left after t years is given y t A( t) 40e, a) What was the amount of Caron 4 left after 000 years? ) How long did it take for the caron 4 to reach 0 grams? c)what is the half-life of caron 4? 6

17 . Suppose $5000 is invested in a certificate of deposit (CD) at 4.75% compounded continuously per year. a) Find the eponential model for this investment, rt A Pe : ) How much will the investment grow to after years? c) When will the investment reach $7500? d) What is the douling time for the investment? 7

18 More applications.. How long will it take a ank. How long will it take a ank. An eponential decay account with $5000 compounded account with $4000 compounded is given y the formula monthly at 5% to reach 40,000? continuously at 8% take to doule? t A 4e. Find the half life. Creating an equation 4. According to U.S. Census Bureau estimates, The population of the United States was 7. million on July, 980, and 49.5 million on July, 990. Let t e the numer of years since 980. A) Find the eponential growth model assuming the a continuous growth rate. What is the growth rate? B) Estimate the population of the US today. Go online and compare. 8