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Math1050 CH06

Exponential and Logarithmic Functions

TermDefinition
For the composite function f(g(x)), the domain includes all solutions to ___ that are in the domain of ___. g(x), f(x)
A fxn is 1 to 1 if no 2 ___ elements are the same. range
T or F? If a horizontal line intersects the graph more than once, it is not 1 to 1. T
T or F? 1 to1 fxn are always either increasing or decreasing. T
The domain of f is the ___ of f inverse (f^-1). range
The ___ of f is the range of f inverse (f^-1). domain
If the fxn of f is a set of ordered pairs (x, y), then the inverse of f is the set of ordered pairs ___. (y, x)
f(f^-1(x))=___ x
___(f(x))=x f^-1
The graph of a 1-1 fxn and its inverse are symmetric with respect to ___. y=x
If y=f(x) then x=___. f(y)
x=f(y) defines f^-1 ___. implicitly
y=f^-1(x) defines f^-1 ___. explicitly
The ___ form of f(x) is the ___ form solved in terms of x. explicit, implicit
What is the implicit form of y=2x+3? x=2y+3
What is the explicit form of y=2x+3? (implicit form is x=2y+3) f^-1(x)=1/2(x-3)
Exponential fxns are in the form f(x)=___. Ca^x
For f(x)=Ca^x, C is the ___. initial value (the y-intercept)
For f(x)=Ca^x, a is the ___ or ___. base, growth factor
For the fxn: (-1,5), (0,2), (1,-1), (2,-4), (3,-7), what is the avg. rate of change? -3
For the fxn: (-1,5), (0,2), (1,-1), (2,-4), (3,-7), is the ratio of consecutive outputs constant? no
For the fxn: (-1,5), (0,2), (1,-1), (2,-4), (3,-7), is it a linear function? yes
For the fxn: (-1,5), (0,2), (1,-1), (2,-4), (3,-7), what is the y-intercept? 2
For the fxn: (-1,5), (0,2), (1,-1), (2,-4), (3,-7), what is the fxn? f(x)=-3x+2
For the fxn: (-1,32), (0,16), (1,8), (2,4), (3,2), is the avg. rate of change constant? no
For the fxn: (-1,32), (0,16), (1,8), (2,4), (3,2), is it linear? no
For the fxn: (-1,32), (0,16), (1,8), (2,4), (3,2), what is the ratio of consecutive outputs? 1/2
For the fxn: (-1,32), (0,16), (1,8), (2,4), (3,2), what is the growth factor (a)? a=1/2 (the ratio of consecutive outputs)
For the fxn: (-1,32), (0,16), (1,8), (2,4), (3,2), is it an exponential fxn? yes
For the fxn: (-1,32), (0,16), (1,8), (2,4), (3,2), what is the initial value (C)? C=16
For the fxn: (-1,32), (0,16), (1,8), (2,4), (3,2), what is the fxn? f(x)=16(1/2)^x
For the fxn: (-1,2), (0,4), (1,7), (2,11), (3,16), is the avg. rate of change constant? no
For the fxn: (-1,2), (0,4), (1,7), (2,11), (3,16), is the ratio of consecutive outputs constant? no
For the fxn: (-1,2), (0,4), (1,7), (2,11), (3,16), is it linear? no
For the fxn: (-1,2), (0,4), (1,7), (2,11), (3,16), is it an exponential fxn? no
For f(x)=a^x, what is the domain? all real numbers
For f(x)=a^x, what is the range? all positive real numbers
For f(x)=a^x, what is the x-intercept? none
For f(x)=a^x, what is the y-intercept? 1
For f(x)=a^x, what is the horizontal asymptote? y=0
For f(x)=a^x, is it 1-1? yes
What is the inverse of y = 2x? y = log_2 (x)
For f(x)=a^x where a>1, is it increasing or decreasing? increasing
For f(x)=a^x where 0<a<1, is it increasing or decreasing? decreasing (NOTE: (1/a)^x = a^-x)
For f(x)=a^x, can a=1? No because the fxn would be a constant function (f(x)=C) instead of an exponential fxn.
For f(x)=a^x, can 'a' be negative? No because it would result in imaginary outputs for rational exponents with even denominators. (i.e., (-4)^1/2=2i) (NOTE: -4^1/2 is NOT = (-4)^1/2, -4^1/2 =(-1)4^1/2)
For f(x)=a^x where a>1, what 3 points will it always contain? (-1,1/a), (0,1), (1,a)
For f(x)=a^x where 0<a<1, what 3 points will it always contain? (-1,1/a), (0,1), (1,a)
The inverse of an exponential fxn is called a ___ fxn. logarithmic
For log x, the domain is ___. all real positive numbers
For log (x+3) the domain is ___. x>-3
Logarithmic fxns are reflections about ___ of its inverse exponential fxn y=x
For log x, the range is ___. all real #s
For log x, the x-int is ___. (0,1)
For log x, the vertical asymptote is ___. x=0
A logarithmic fxn is increasing if ___. a>1
A logarithmic fxn is decreasing if ___. 0<a<1
For log x, what 3 points does its graph always contain? (1/a,-1), (1,0), (a,1)
log 1 = ___ 0
log_a a = ___ 1
log 10 = ___ 1
log_a a^r = ___ r
log 10^2 2
a^(log_a M) = ___ M (because log_a M = x and a^x = M. So replacing x in a^x = M with log_a M give us a^(log_a M) = M)
If f(x) = a^x, then f inverse = ___ log_a x
log M + log N = ___ log MN
log M - log N = ___ log M/N
log M^r = ___ rlog M
a^x = e^___ = 10^ ___ xln a, xlog a
T or F? log M + log N = log (M + N) F. It = log MN
T or F? log M - log N = log (M - N) F. It = log M/N
T or F? rlog M = (log M)^r F. It = log M^r
What is the change of base formula? log_a M = (log M)/(log a) or (ln M)/(ln a)
What solutions to logarithmic equations are extraneous? domains that are negative
When solving logarithmic equations it is important to avoid using the property log x^r = rlog x for ___ values of r. even
If a^u = a^v then ___. u=v
If log u = log v then ___. u=v
What is the simple interest formula? I=Prt
What is the compound interest formula? A = P(1 + r/n)^nt
What is the continuously compounding interest formula? A = Pe^rt
The equivalent annual simple interest rate that would yield the same amount as compounding is called the ___. effective rate
What is the formula for effective rate (r_e) of interest for compounding, but not continuous? r_e = (1 + r/n)^(n -1)
What is the formula for effective rate (r_e) of interest for compounding continuously? r_e = (e^r) -1
The amount a value of money will become after an investment is called the ___. (i.e., what a present value will be worth) present value
What is the present value formula for compounding interest, but not continuously? P = A/(1 + r/n)^nt or P = A(1 + r/n)^-nt
What is the present value formula for continuously compounding interest? P = A/(e^rt) or P = Ae^-rt
Uninhibited growth and decay follow the function ___ where A_0 is the ___, k is a ___ that is always ___ for growth and always ___ for decay, and t is ___. A(t) = A_0e^kt, starting value (pronounced A-naught where t=0), constant, +, -, time
What is the formula for the law of uninhibited growth? N(t) = N_0e^kt
For N(t) = N_0e^kt, N_0 = ___. pronounced N-naught, the initial value (the number you start with)
For N(t) = N_0e^kt, k = ___. the growth rate (always a positive constant)
For uninhibited growth, k is always ___. For decay, it is always ___. +, -
For the logistical model of population P = c/(1 + ae^-bt), a, b, and c are ___. c is the ___. b is the ___. constants, carrying capacity, growth rate (k)
The inflection point is ___. half of the carrying capacity
T or F? y = 2^-x is the same as y = (1/2)^x T
For A = P(1 + r/n)^nt, A = ___. amount earned after the end of the term
For A = P(1 + r/n)^nt, P = ___. beginning principal
For A = P(1 + r/n)^nt, r = ___. interest rate in decimal form
For A = P(1 + r/n)^nt, n = ___. the number of times a year it compounds (quarterly = 4, semiannually = 2, biannually = 1/2, etc.)
For A = P(1 + r/n)^nt, t = ___. time in years of the term
T or F? For growth and decay formulas, t is in whatever time period the constant was figured in. (i.e., days, weeks, years, etc.) True
D of a logarithmic fxn is always (+) because the log of a (-) # is ___ since there are no ___ solutions that will result in a (-)x. undefined, even
What is the inverse of y = log_2 (x)? y = 2x
y = log_2 (x) is a reflection about ___ of y = ___. y=x, 2x
T or F? ln (x) = log_e (x). True
Exponential fxns have a ___ asymptote and logarithmic fxns have ___ asymptote. horizontal, vertical
Created by: drjolley