Test 2 (functions, exp, log, proofs) Flashcards
Unit
values that we add to an operation that does not change the result
Injective in other words
one-to-one
Surjective in other words
onto
Identity function
y equals x
Reciprocal function
1 over x
Inverse square function
1 over x squared
What is the number in a logarithm called
argument
Which transformations are rigid
translations, reflections and rotations
Which transformations are non-rigid
stretches and shrinks
Vieta’s formulas
sum of roots equals -b/a; product of roots equals c/a
Fundamental theorem of algebra
every nonconstant polynomial has a complex root (therefore, every polynomal of degree at least 1 can be factored to linear factors)
Polynomials theorem 1
integer root divides constant coefficient of a polynomial
Polynomials theorem 2
polynomials have a root a / b, where a divides a0 and b divides an
Degree of a polynomial root
how many times it appears
Division theorem for polynomials
for dividing every polynomial by another polynomial there exist two other polynomials, so that one is a factor and the other the remainder
Remainder theorem
remainder when dividing a polynomial by x - a is equal to p(a)
Factor theorem
a polynomial has a factor (ax - b) iff p(b/a) equals 0
Vieta’s formulas for general polynomials
sum of root equals -(an-1)/an; product of roots equals (a0/an)*(-1)**n
Rational function
function of the form of the quotient of two polynomials
When is a rational function in a reduced form
if both polynomials have no common factor
What happens if we have a pole of an odd degree
graph will change sign
Where does a rational function intersect an asymptote
at the roots of the remainder
How is a curved asymptote called?
oblique asymptote
When do we have to check the solution of an equation?
if we squared it at any point; we used the rules of logarithms
What is half-life?
time it takes for a given amount of material to decrease to half of its original amount
When does the inequality sign change?
if we put both sides as the exponent of a function with base < 1
What is a statement?
A sentence that can be true or false
What is an axiom?
an original statement, which we want to hold, to define more complex statements from it
Which words tell you of implication?
if A then B
A implies B
from A follows b
A only if B
Which words tell you of equivalence?
A iff B
A is equivalent to B
What are antedecent and consequent?
in implication A is antedecent and B is consequent
What is tautology?
a statement that is always true
What is the contrapositive of ‘if P then Q’?
‘if not Q then not P’, they are equivalent
What is the converse of ‘if P then Q’?
‘If Q then P’, they are not equivalent
What is equivalence ‘P iff Q’ same as?
‘If P then Q’ and ‘If Q then P
