Alg Functions Algebra Assignment Sheet Functions, Fog, Gof, Inverse, Logs ( Assignment # Functions, Domains ( Assignment # Composition of Functions ( Assignment # Inverse Functions ( Review Sheet (5 Quiz (6 Assignment # Eponential Equations (7 Assignment # 5 Logarithms (8 Assignment # 6 Laws of Logarithms (9 Assignment # 7 Laws of Logarithms (0 Assignment # 8 Calculator Logarithms ( Review Sheet ( TEST
Alg Functions Functions I Vocabulary a Function: The set of (, y pairs such that b domain: c range: Vertical Line Test II Functions? Y/N Domain A. a (, (8, (8,8 b (,- (, (5,- c y + 5+ 6 d y e y 8 f y g y -( + h f ( f( is the same as y
Alg Functions III Restrictions Denominator 0 5 +, - -5, + 5-6 for even roots: negative number Find the domain of the following please. a f ( b f ( + 5 c f ( d y e y 6 + 8+ 5
Alg Functions IV Evaluating functions for y when given a value of Evaluating: Substituting in numbers y f( y + + f ( + + f ( + + f ( + ( + f ( ( ( + + f ( + + f(- f(0 f g( + Find g( + If f( + g( - 5 a find f( + b find f( g( c find f(g(
Alg Functions 5 ( ( Algebra Assignment #, ( f 6 7 8 g + ( (a g( (c f g(. Find each of the following. (b f( 6 (d The remainder when f( is divided by ( i ( Determine whether each of the following defines y as a function of. If it is a function, find the domain please. (a y 5 (f y 5 (b y + (g y 5 6 (c + y (h y + (d y + 5 (i y 5 + (e y + 8 7 + 6 (j y 9 + 5 ( Let the function f be defined by f( +. If f ( ϑ 0, find ( f ϑ. ( Let the function f be defined by f( + c, where c is constant. If ( constant c. f 0, find the value of the
Alg Functions 6 Answers ( (a 0 (c 7 (b (d i ( (a Real (f 5 (b Not a function (g or 6 (c Not a function (h Real (d (i 5 (e,, (j or 5 ( 8 ( 8
Alg Functions 7 Composition of Functions EXAMPLES Functions can be added, subtracted, multiplied and divided.. If f ( and g( find a f( + g( b f( g( c f( g( d f( g( e f( g( Notes: Let f ( and g( f g ( f(g( Use composition to find the following: If f ( and g( find a f ( g( b g( f ( If f (, g( and h( find f g h(
Alg Functions 8 6 + If f ( and g( find f g +5 DOMAIN: ( ( f + 8 + 5, and g +. If f ( and g( find the domain of g f Find all values of such that ( ( ( ( f g g f if
Alg Functions 9 Algebra Assignment # Composition of Functions ( Find each of the following numbers, given the functions below. f ( ; g( ; h( + (a f ( h( (b g ( h( (c g ( f( 5 (d h ( f( g( ( Find f ( g( and ( f( g for each of the following please. (a f g ( + ( 5 (b f g ( ( + 5 + ( Find g ( if f( + 5, and f( g( + 9. ( Find f ( f( if f( + (5 Find all values of such that f ( g( g( f( if f( +, and g(.
Alg Functions 0 Answers ( (a (b (c 89 (d 6 ( (a f( g( 8 + 6 (b f( g( 7 5 + g ( f( + 8 9 g( f( 0 + 8 ( g( + ( ( f( f (5
Alg Functions Inverse functions ( NOTES: f and f f is not f A. If f( and g( are inverses of each other then f(g( and g(f( (they must be one to one to be inverses E. If f( - g( g( + + + f + f( g[ ] ( +
Alg Functions If f ( and g(, Prove f( and g( are inverses of each other B. Finding inverses Switch the domain and the range y Then solve for y EX. f( - 5 y 5 switches to y 5 + 5 solve y this is the inverse E. Find f if f( Domain restrictions become:
Alg Functions V Practice Find f ( for each of the following: a f( + b f( ( + c f( 5 Find ( ( ( ( f g and g f for each of the following: e ( ( f g + 6 f ( ( f + g + Find each of the following numbers, given the functions below: ( ( ( f +, g +, h g ( ( ( f h h g h( j h( f( 0
Alg Functions Algebra Assignment # Inverse Functions ( Find f ( for each of the following please. (a f ( 5 + (e f( 5 7 (b f ( (f f ( + 5 + (c f( 5 + (g ( f + 5 (d f( 7 + 7 (h f( 5 + 8 ( f( 6 + + 5, g ( f(. Find ( g. ( f(. Find f (, and sketch a graph of f ( and f ( on the same set of aes.
Alg Functions 5 Answers ( (a f ( 5 (e f ( + 5 7 (b f ( (f f ( (c f ( 5 + (g f ( 5 (d f ( 7 + 7 (h f ( ( + 5 + 8 ( g ( + 5 6
Alg Functions 6 Algebra Review Worksheet ( Find each of the following numbers, given the functions below. f ; ( ( g ; h ( + (a f ( h( (b g ( h( 0 (c f ( h( g( 8 (d g ( f( h( 8 ( Find the domain of each of the following functions please. f 9 (a ( f (b ( 9 5 + (c f( (d ( + + 6 f ( + ( ( 5 ( Find f ( g( and ( f( g for each of the following. (a f g ( + ( (b f g ( ( + + ( Find f ( for each of the following. (a f( 5 7 (b f( + 5 (c f( + 5 (d f ( 5+ (5 Find all values of for which f ( g( g( f( if f( 5 and g( +. (6 Find g (, if f( + + and f( g( 6 +.
Alg Functions 7 Answers ( (a 0 (b (c 5 (d 9 ( (a or (b and 8 (c,, (d or or > 5 ( (a f( g( 5, g( f( + 8 + 5 (b f( g(, g( f( ( (a ( 7 f + 5 (c f ( 5 + 5 + + 8 (b f ( ( + 5 f 5 (d ( (5 5 (6 g(
Alg Functions 8 Eponents I Review y b b 0 b ( b y b b y b b c b b ( b b b II y If b b, then y If b a, then b a E. 5 + E. + 5 6 + 8 6 6 5 7 + 9 + 5 6 8
Alg Functions 9 7. + + 79 8. 5 + 96 + 9. + + + 5 III Sketching y y NO negative bases - (0, on every graph b - - 0 y - - - 0 y
Alg Functions 0 Algebra Assignment # Eponential Equations ( Solve for please. (a + 8 + (b 5 5 (c 7 8 (d 9 7 + (e ( 8 6 (f ( 6 (g 7 8 (h 8 0 (i 7 0 7 + 9 0 (j 6 0 + 6 0 ( Sketch a graph of each of the following. (a y (b y
Alg Functions Answers ( (a (b 5 (c 0 (d, (e 9 0 (f ± (g 9 (h (i 0, (j,
Alg Functions EXPONENTIAL EQUATIONS EXTRA ( Evaluate each of the following numbers please. (a 9 i 9 (c i (b + (d ( Solve each of the following equations please. (a 5 (e ( 5 + 7 8 (b 5 8 (f ( 6 + 9 7 (c + + 8 6 (g 8 6 i 8 + 8 0 (d ( + 8 6 i (h 9 i + 8 0 ( Sketch a graph of each of the following on the same set of aes. y and y log (
Alg Functions Answers ( (a 7 7 (c (b 7 (d 6 ( (a 5 (e ± 5 (b (f, 6 (c 7 (g, (d 8 (h 0,
Alg Functions LOGS Logs are the inverse of eponential functions. If y + to find inverse y+ y If y to find inverse How do you solve for y? y Logarithms are eponents and follow eponent rules. EXPONENTIAL FORM LOGARITHM FORM p b base power n number number log n p b base power 7 log 7 5 log5 5 5 8 log 8 log RULES
Alg Functions 5 EXAMPLES log 8 log y log 5 z 5 log 6 p 8 log c 8 log 9 d log 7 ( log 9 log 6 a 6 log 5 warm-up before net lesson - - - 5 + i 6 ( 8 9 i + 8 0 8 log 7 log - 7 5 b log (log 8 6 7 7 log.000 y
Alg Functions 6 Algebra Assignment # 5 Definition of the logarithm ( ( 6 Solve for please. log (0 5ln( e log 6 ( ( ( ( log ( log ( 9 ( log 6 ( 6 5 ( log ( ( log ( log 5 ( ( (5 ( 5 log 8 (6 ( log (5 log ( 8 6 ln log 7 (6 ( e ln ( e (7 ( 8 + 5 log 7 (7 log log ( (8 ( 7 ( (9 log ( 6 ( log log (8 ( 9 6
Alg Functions 7 Answers ( (0 0 ( 6 ( 5 ( ( ( 9 ( 8 (5 ( 6 5 (6 (5 6 (7 (6 (8 (7 (9 8 (8
Alg Functions 8 LAWS OF LOGS Properties of Logs I LAWS log mn log m+ log n b b b m logb logbmlog bn n II Equations If log m log n then m n b ( log log b If log m n then b m b (log number n log m b p plog m b III Eamples log ( n + 0 log ( 5n 5 5 log6 8 log6 w log6 log + log 7 log log m log 8 5 0 0 log m log 6 log 6 7 7 + 7
Alg Functions 9 log0 m+ log0 m log0 8 log6 log6 8 log6 7 ( log ( log ( 9 + + 0 log ( y+ log ( y log log 5 log 7 log ( + + log ( log 7
Alg Functions 0 Algebra Assignment # 6 Logarithmic Equations ( Evaluate each of the following please. log (a 7( 6 7 (b log5( 6 5 (c log8( 7 log( 5 + 8 (d e ( ( ln 8 ln ( Solve for please. + (b log ( (a log( log( 6 0 + 9 log log (d log ( log ( log ( 8 (c ( ( 5 5 9 9 9 (e log ( 7 + log ( log ( 6 (f ln( 5 + ln( ln( 05 ln( (g log ( + log ( + log ( 7 (h log ( + + log ( log ( 6 0 0 7 (i log ( + log ( log ( 6 + (j log ( + + log ( log ( 8 8 8 5 8 8 (k log ( log ( (l log( 9 log9( 5 8
Alg Functions Answers ( (a 6 (b 6 (c (d ( (a 7 (b 0, (c 9 (d 6 (e (f (g (h 5 (i (j (k, 5 (l
Alg Functions Algebra Assignment # 7 Logarithmic Equations Solve for please. 5 ( log ( log ( ( log ( log ( log ( 5 6 6 6 ( ln( + + ln( + ln( 9 9 7 8 8 ( log ( log ( (5 ln( + + ln( ln( 7 8 ln( log9( 5 + + 5 e log log (6 ( ( (7 log ( 5 log ( + 7 (8 log ( 9 ( log ( 0 (9 ( 8 8 ( log 5log ( + 0 (0 ln(
Alg Functions Answers (, 7 (,, ( reject 5 (,, ( reject (, (reject (5, ( reject, (6 7, (reject (7 9, (reject (8 8, (9, (0 Ø
Alg Functions Algebra Assignment # 8 Calculator Logarithm Problems ( Use a calculator to solve each of the following correct to decimal places please. (a 5 0 (b + 9 (c log ( 8 (d log ( (e ln (. (f 7. log ( 5 (g 0. > 7 (h ln( ( ln( 5 0 ( Let log 0( p and ( log 0 q. Evaluate each of the following in terms of p and q. (a log 0( 6 (b log 0( 7 (c log 0 5 6 (d log 0( 90 (e log 0( 0.5 (f log 0( 5 ( Simplify the following epression please. ( 5 i ( i ( 7 log log log 9 5 ( The magnitude of an earthquake is measured using the Richter scale; M E log 0., Where M is the magnitude of the earthquake, and E is the seismic energy released by the earthquake (in 5. 0 joules. Calculate the joules. The 989 San Francisco earthquake released approimately magnitude of the earthquake using the Richter scale. How much energy would be released (in joules by an earthquake which measures 8. on the Richter scale?
Alg Functions 5 Answers ( (a.86 (b 0.76 (c.609 (d 6.8 (e.87 (f 5.88 (g <.66 (h 0.6065, 0.0855 ( (a p + q (b p + q (c q p (d q + 5 (e p (f p ( 5 8 ( 7., 6 7.079 0 joules
Alg Functions 6 Algebra Review Worksheet ( Solve for please. (a + 9 7 (i log9( 9 (b 5 + 8 6 (j log9( log( 9 (c ( 8 i (k ( (d 8( 6 log ( log ( 0 8 8 log (l log ( + log ( 5 5 (e log ( (m ( + ( log + log 5 (f log ( 6 (n log ( log ( + log ( 8 (g log (.5 (o log7( + + log7( + log7( + log7( 0 ( log log (p (h ( 8 ( log( 6 ( log log + 8 ( Use a calculator to solve for. Epress answers correct to decimal places. (a 8 (b 5 (c log( (d ln( ( ln( 0
Alg Functions 7 Answers ( (a 8 (i (b 9 (j (c (k 8, (d 9 (l (e (m (f 8 (n (g (o (h (p ( (a.89 (b 0.8 (c 0.6 (d, 7.89, 0.5
Alg Functions 8 Algebra /Trig Etra Optional Review B. Solve for :. + 9 7. ( + 5 8. 8 6. log8 5. log 6. log log9 7. log (0.5 8. log (log8 9. 9 log log 9 0. 9. (log log 0 8 8. log 5(+ log 5(. log ( + log ( log. log ( log (+ log8 5. log (+ log log log log 6 6. + 8 7. log (+ + log + log (+ log 0 7 7 7 7 C. Isolate completely.. 8. 5. 5 + +
Alg Functions 9 ANSWERS B.. 8.. -. 9 5. 6. 7. 8. 9. 0. -.,8... No solution 5. 6. 7. (the other are imaginary C. log8.89 log. log0.8 log 0. log 5.759 5 log