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# Math1050 CH06

### Exponential and Logarithmic Functions

Term | Definition |
---|---|

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 |