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Math 70 12.1 Algebra of Functions

Overview of Chapter 12 12.1 Function Composition

12.2 12.3 12.4 12.5 12.6& 12.7

Inverse Functions [Functions that "un-do" each other when composed] Exponential Functions Logarithmic Functions [Inverse functions (12.1) of Exponential Functions (12.2)] Properties of Logarithms [Unexpected traits of Logarithmic functions (12.3)] Natural logs, and Change of Base [Special logarithmic functions(12.3) and GC] Solving Exponential and Logarithmic Equations and Applications

*This chapter builds one section on the next, layering complicated concepts.

Objectives 1) Find a new function which is composition(/ o g)(x) of two given functions.

2) Review 5.9: the sum(/+ g)(x), difference(/ - g)(x), product(/· g)(x), quotient(/ I g)(x)

3) Recognize function notation and notation for the names of these functions. 4) Practice negative and positive exponents on common bases, in preparation for 12.2.

Practice and Examples

1) Given /(x)=3x2 +4x+land g(x)=2x-5,find: ~ a) (/ + g)(x)

~

b) (/-g)(x)

c) (/ · g)(x)

d) (/I g)(x) e} (go /)(x)

f) (/ o g)(x)

/(-1)=4

2) Given /(0)=5

/(2)=7

/(7)=1

#' a) (f + g)(2) b) (/-g)(O)

c) (/ · g)(7)

v d) (/ · g)(O)

e) (/I g)(O) y f) (gl /)(0) .,/ g) (go /)(2)

< h) (f 0 g)(2)

g(-1) =-4

g(0)=-3 ,find

g(2) =-1

g(7) =9

-sLlp 3) On interstate trips, a driver averages 54 mph. The distanced in miles traveled int hours is given by

d(t) = 54t. Because the driver averages 25 miles per gallon, the number of gallons g used is given by g( d) = d I 25. The cost per gallon is $2.95, so the total fuet cost is given by c(g) = 2.95 g.

a) Write a function describing the number of gallons used int hours of travel. b) Write a function describing the total fuel cost int hours of travel. c) Determine the total fuel cost of a 12-hourtrip.

~It; p

4) An oil tanker runs aground and springs a leak. The oil spreads out in a semicircular pattern from the shoreline. The distance r (in feet) from the tanker to the edge of the oil spill at time t (in minutes) is given by the function r(t) = 20t .

a) If the area of the semicircle is given by A(x) = .!.nx2, wherex is the radius, write a function for

2 the area covered by the oil at time t

b) What is the area of the oil spill after 5 minutes?

'i fl$

5) Given /(x)=2x+3and g(x)=.!.(x-3), find: 2

a) (go f)(x)

b) (/ ogXx)

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h) l~o3)(t<) ~ 1-1

On interstate trip~ a driver averages 54 mph. The distance d in miles traveled in t hours is given by d~t) =t:54t. Because the driver averages 25 miles per gallo~ the number of gallons g used is given by

g(d) ~ 'd I 25. The cost per gallon is $2.9~, so the total fuel cost is given by c(g) = 2.95g. a) Write a function describing the number of gallons used int hours of travel. b) Write a function describing the total fuel cost int hours of travel. · c) Determine the total fuel cost of a 12-hour trip.

o-) @Cd [-b)) := Si./i: :='.-Ei&t J . ~

b) cC°o-Crllt)))::. 2.15 ( ~. 10 -1:;) ~ J0.?7d- t ~I

c} c(~,.{d[t)))-::: 'l0.-,'7d-)(1g.) =- ~1-6.Y~Y __

x LM~Yto 1 An oil tanker sµns aground and springs a leak. The oil spreads out in a semicircular pattern from the shoreline. The distance r (in feet) from the tanker to the edge of the oil spill at time t (in minutes) is given by the function r(t) = 20t .

a) If the area of the semicircle is given by A(x) = .!..11X2, where xis the radius, write a function forthe 2 ,

area covered by the oil at time t. b) What is the area of the oil spill after 5 minutes?

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Math 70 Practice with Exponents

~' Complete the table by raising each base to each exponent. Some example entries are provided.

Exponents across-+ -3 -2 -1 0 2 3 Bases down!

-} ~ o-3 {\A~~ tr ~ \ W\~~{\~ 1 J_ ' oo 0 n-' ~ IAvtJe-P,~ 0 0 indeterminate

1 [l JU OJ I I ) 2

-3 ' ~ -

i-1 = _!_ ~_!_ -7-- \ J_ I ·~ ~· ~-:;. ~~ ~ ~ ~ -- l-\ 21 2 ,,

3 ~) j_ ~ \ J_' 51 ;:::- (\ \ 3°=1 '3 -:;;. '3 ~ ::::- . -;;} 'I D\ I 9 '1}

4 .. -, ~;: ~\ \-1-\..\ "'.'- \I.:. ~ I \~ "~

5 ... , ..L -\J-~ ' ft\ I 52 =25 l2S-; -::. -s" - ' '2.-S- ;)._"!)'

6 _, \@ ,_-_L

\{\ '

~b 63 = 216 lo :;.. ~ "'.:--· ";;:\ /,,. l '} (,,. \

-1 <-1t3 = <-~>3 =ED

-1- ')... -1 ' / '2 3>

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"l-

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'2

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