Formulas - Evaluation 3

Question 1

Remainder theorem

Answer 1

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)

Question 2

Factor Theorem

Answer 2

let f be a polynomial function, (x-c) is a factor of f(x) if f(c) = 0

Question 3

Intermediate value theorem

Answer 3

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

Question 4

Composite function notation

Answer 4

fog(x) is f(g(x))

Question 5

Finding the domain of a composite function

Answer 5

find the domain of f(g(x))

1. find the domain of f(x)
2. 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)

Question 6

Composite function on a graph

Answer 6

f(g(-1))

1. find g(-1)
2. plus the value for g(-1) into f(-1)
3. f(-1) is the answer

Question 7

Inverse function conditions

Answer 7

function must be 1-1 for it to have an inverse

f(x) becomes f⁻¹(x)

Question 8

Inverse function graphically

Answer 8

reflect graph in line of y = x

Question 9

Inverse of a function algebraically

Answer 9

1. set function as y equals..
2. swap x and y
3. solve for y
4. f⁻¹(x) = ____

Question 10

Domain and range of inverse function

Answer 10

range and domain swaps for f(x) and f⁻¹(x)

Question 11

Power function vs exponential function

Answer 11

power: y = x²
exponential: y = 2ˣ

Question 12

Exponential rules

Answer 12

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

Question 13

Exponential function notation

Answer 13

f(x) = Caˣ
C : initial value because f(0) = Ca⁰ = C
a : growth factor (if positive)

Question 14

Solving exponential equations

Answer 14

set both bases to the same then solve normally

Question 15

e

Answer 15

e is the irrational number that approximately equates to 2.7

Question 16

Log function definition

Answer 16

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

Question 17

Log base 1

Answer 17

log₁x=y
1ʸ = x
x = 1

Question 18

Graph of logₐx

Answer 18

exponential function graph of y=aˣ
D: (-∞,∞)
R: (0,∞)
reverse for y=logₐx

Question 19

Graphical transformations with log

Answer 19

basic function: y = log₂x

regular order of transformations, just transform log graph

Question 20

Log functions in relation to base

Answer 20

the base does not impact the domain (x)
x must strictly be positive and not equal to zero

Question 21

Solving log equations

Answer 21

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)

Question 22

log to ln

Answer 22

logₑx = lnx

log(x) = log₁₀(x)

log₂ = C , 10ᶜ = 2

ln3 = a, eᵃ = 3

Question 23

log base rule

Answer 23

logₐm = logₐn

m = n

Question 24

log properties

Answer 24

logₐ1 = 0
logₐa = 1
logₐaⁿ = n
aˡᵒᵍˣa = x
logₐ(mn) = logₐm + logₐn
logₐ(m/n) = logₐm - logₐn
logₐxⁿ = nlogₐx
eʳˡⁿᵃ = aʳ

Question 25

Change of base rule

Answer 25

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

Question 26

Logarithmic and exponential functions

Answer 26

x is an accepted solution so long as it fits the x>0 requirements on both sides of equation (domain on both sides)

Question 27

Condition to apply log properties

Answer 27

needs same base to apply rule

Note:
logₑ = ln
log₁₀ = log

Question 28

Solving exponential equations

Answer 28

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

Question 29

Rule for domain of exponential function

Answer 29

for an exponential function (given a>0 and a≠1) the domain is the domain of the exponent

Question 30

Exponential growth/decay models

Answer 30

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

Question 31

When to use log vs ln

Answer 31

take ln when there is an e or different bases

take log when there is the same base