Formulas - Evaluation 3 Flashcards
Remainder theorem
f(x)/g(x) = q(x) [ R(x)/g(x) ]
f(x) - dividend
g(x) - divisor
q(x) - quotient
R(x) - remainder
if f(x) is divided by (x-c), the remainder is f(c)
Factor Theorem
let f be a polynomial function, (x-c) is a factor of f(x) if f(c) = 0
Intermediate value theorem
if f(a) < f(b) & f(a) and f(b) are opposite signs then there is at least one real zero of f between a and b
Composite function notation
fog(x) is f(g(x))
Finding the domain of a composite function
find the domain of f(g(x))
- find the domain of f(x)
- find the domain of g(x)
Domain is all set of x except what is not included in the domain of g(x) and the value of g(x) that makes what is not included in the value of f(x)
Composite function on a graph
f(g(-1))
- find g(-1)
- plus the value for g(-1) into f(-1)
- f(-1) is the answer
Inverse function conditions
function must be 1-1 for it to have an inverse
f(x) becomes f⁻¹(x)
Inverse function graphically
reflect graph in line of y = x
Inverse of a function algebraically
- set function as y equals..
- swap x and y
- solve for y
- f⁻¹(x) = ____
Domain and range of inverse function
range and domain swaps for f(x) and f⁻¹(x)
Power function vs exponential function
power: y = x²
exponential: y = 2ˣ
Exponential rules
aᵐ.aⁿ = aᵐⁿ
aᵐ/aⁿ = aᵐ⁻ⁿ
(aᵐ)ⁿ = aᵐⁿ
(a/b)ⁿ = aⁿ/bⁿ
(ab)ⁿ = aⁿbⁿ
1/aⁿ = a⁻ⁿ or 1/a⁻ⁿ = aⁿ
1ⁿ = 1
a⁰ = 1 a ≠ 0
Exponential function notation
f(x) = Caˣ
C : initial value because f(0) = Ca⁰ = C
a : growth factor (if positive)
Solving exponential equations
set both bases to the same then solve normally
e
e is the irrational number that approximately equates to 2.7
Log function definition
the logarithmic function with base a (a≠1 and a>0), is denoted by y = logₐx and defined by y=logₐx if and only if
x = aʸ
Log base 1
log₁x=y
1ʸ = x
x = 1
Graph of logₐx
exponential function graph of y=aˣ
D: (-∞,∞)
R: (0,∞)
reverse for y=logₐx
Graphical transformations with log
basic function: y = log₂x
regular order of transformations, just transform log graph
Log functions in relation to base
the base does not impact the domain (x)
x must strictly be positive and not equal to zero
Solving log equations
f(x) = logₐ(___)
(___) must be greater than zero
find zeros from numerator
asymptote from denominator
only solutions that are accepted are those that are positive or within positive range as determined by numberline (which also determined range)
log to ln
logₑx = lnx
log(x) = log₁₀(x)
log₂ = C , 10ᶜ = 2
ln3 = a, eᵃ = 3
log base rule
logₐm = logₐn
m = n
log properties
logₐ1 = 0
logₐa = 1
logₐaⁿ = n
aˡᵒᵍˣa = x
logₐ(mn) = logₐm + logₐn
logₐ(m/n) = logₐm - logₐn
logₐxⁿ = nlogₐx
eʳˡⁿᵃ = aʳ
Change of base rule
logₐ= ln x / ln a
logₐx = log x / log a
ex
log₃5 = ln 5 / ln 3 = log 5 / log 3
log√₂ √5 = ln√5 / ln√2 = ln 5 / ln 2
Logarithmic and exponential functions
x is an accepted solution so long as it fits the x>0 requirements on both sides of equation (domain on both sides)
Condition to apply log properties
needs same base to apply rule
Note:
logₑ = ln
log₁₀ = log
Solving exponential equations
Term with exponent must be in simplest form before solving
for anything that’s _²ˣ
substitute y = 2ˣ
then solve for 2ˣ with inclusion of domain
for double rational functions check range against condition must be greater than 0 and number line to find domain
Always state domain
Rule for domain of exponential function
for an exponential function (given a>0 and a≠1) the domain is the domain of the exponent
Exponential growth/decay models
A(t) = A₀eᵏᵗ
A₀ = initial value
k = growth/decay rate (growth if k >0, decay if k<0)
t = time (units dependant on question)
- solve taking natural log
When to use log vs ln
take ln when there is an e or different bases
take log when there is the same base
