Maths - Semester 1

Question 1

What are the components of a set?

Answer 1

A list of objects that doesn’t repeat, with each object in a set called an “Element”

Question 2

How does a function relate sets?

Answer 2

It acts as a rule that assigns an element in a domain set D to an element in a range set E

Question 3

What is a function composition in terms of sets?

Answer 3

When the range of one function is used as the domain of another.

Question 4

How do graphs work in set theory?

Answer 4

A graph in set theory is a set of ordered pairs consisting of a domain and range value from a function. These ordered pairs are plotted with the y value as the range value.

Question 5

What is a formal function?

Answer 5

A formal function is one where each domain value is linked to only 1 range value. If a line in the x axis intersects the graph twice, it shows that it has two range values and the function is not formal.

Question 6

What is a piecewise function?

Answer 6

A piecewise function is one which is broken into multiple sections, with each section having a different rule relating the domain and range.

Question 7

What is an even function?

Answer 7

An even function is one where a negative domain value -x has the same range value as its positive counterpart. In graph form this causes a y-axis reflection.

Question 8

What is an odd function?

Answer 8

An odd function is one where the positive range value for a negative domain value -x is equal to the negative range value for a positive domain value x. In graph form this looks like it is symmetrical rotated 180 degrees.
f(-x)=-f(x)

Question 9

What makes a function increasing/decreasing?

Answer 9

A function is increasing if the range values go up as the domain values go up.
A function is decreasing if the range values go down as the domain values go up.

Question 10

How would you stretch or shrink a graph vertically by a factor of C?

Answer 10

y = cf(x)

Question 11

How would you stretch or shrink a graph horizontally by a factor of C?

Answer 11

y = f(cx)

Question 12

How would you reflect a graph in the x axis?

Answer 12

y = -f(x)

Question 13

How would you flip a graph in the y axis?

Answer 13

y = f(-x)

Question 14

How do you move a graph up or down?

Answer 14

y = f(x) + c

Question 15

How would you move a graph left or right?

Answer 15

y = f(x+n)

Question 16

Is the product of two even functions even?

Answer 16

Yes

Question 17

Is the product of two odd functions odd?

Answer 17

No, it is even.

Question 18

What is the definition and formula of a polynomial?

Answer 18

A polynomial is a function consisting of variables and coefficients.
It has a basic formula of:
P(x) = anx^n + a(n-1)x^(n-1) + … + a1x + a0

Question 19

When is the degree of a polynomial n?

Answer 19

When the coefficient a(n) is not 0.

Question 20

What is the form of a power function?

Answer 20

f(x) = x^a

Question 21

What are the special cases of the power function?

Answer 21

When a is an integer n, the power function is a polynomial with 1 term.
When a is a number 1/n, then the power function is a root function.
When a is -1, then the power function is a reciprocal function.

Question 22

What are trigonomic functions?

Answer 22

They are periodic functions that repeat every 2*pi units, and have a range of closed interval [-1,1].

Question 23

What are all the trigonomic functions and their derivations from sin(x) and cos(x)?

Answer 23

Sin(x) = Cos(x + pi)
Cos(x) = Sin(x + pi)
tan(x) = sin(x)/cos(x)
Cosec(c) = 1/sin(x)
Sec(x) = 1/cos(x)
Cot(x) = cos(x)/sin(x)

Question 24

What is the form of an exponential function?

Answer 24

f(x) = a^x

Question 25

What is the form of a logarithmic function?

Answer 25

f(x) = logb(x)

Question 26

What is the inverse of an exponential function?

Answer 26

A logarithmic function

Question 27

What is the behavioural of an exponential function when the base is greater than 0 but less than 1?

Answer 27

It is a decreasing function.

Question 28

What is the behaviour of an exponential function when the base is greater than 1?

Answer 28

It is an increasing function

Question 29

What makes two sets equal?

Answer 29

If they have exactly the same elements

Question 30

Does the order of a set matter?

Answer 30

No

Question 31

What happens if the same element appears multiple times in a set?

Answer 31

Any repeat elements are ignored and removed

Question 32

How can one set be a subset of another?

Answer 32

If one set contains all the elements of another.

Question 33

What is the empty set?

Answer 33

A set that contains no elements

Question 34

What is the meaning of P⇒Q?

Answer 34

It means "P implies Q", and states that if a statement P is true, then the statement Q must also be true

Question 35

What is a Negation Statement?

Answer 35

A negation statement is the mathematical opposite of an original statement.

Question 36

What are the 3 different ways of saying "P implies Q"?

Answer 36

"If P then Q" "Q if P" "P only if Q"

Question 37

How does a negation statement impact an implication statement?

Answer 37

If a statement P relies on another statement Q to be true, then "not Q" (the negation of Q) implies "not P" (the negation of P). In other words, if P relies on Q, then Q not happening means P wont either.

Question 38

What is the symbol "∃" and what does it mean? How can it be treated when proving a statement?

Answer 38

The symbol "∃" is the existence quantifier and means "there exists". Statements involving it can be proved by finding one example where the statement is true.

Question 39

What is the symbol "∀" and what does it mean? How can it be treated when proving a statement?

Answer 39

The symbol "∀" is the universal quantifier and is short for "for all". Statements involving it can be proved by providing a general argument that is true for all cases.

Question 40

How are the universal and existence quantifiers related?

Answer 40

They are negations of one another, as to prove one false required the other. In a "for all", there must exist one example when the rule is false and vice-versa.

Question 41

What does it mean if a statement starts as "∀x∈∅"?

Answer 41

It means the statement is automatically true, as there are no elements in the empty set meaning all elements in the set share any rule.

Question 42

What does it mean if a statement starts as "∃x∈∅"?

Answer 42

It means the statement is automatically false, as there are no elements in the empty set meaning there doesn't exist an element for a certain rule to be true.

Question 43

What is a tangent and a secant line?

Answer 43

A tangent line is one which touches the curve, and continues in the curve direction at the point of contact. A secant line is one which intersects the curve more than once.

Question 44

How would you write a limit that is approached from the left and right hand sides?

Answer 44

If approached from the left, the a value will have a "-" symbol. If approached from the right, the a value will have a "+" symbol.

Question 45

How can it be proven that a limit exists?

Answer 45

Check if the limit is the same when approached from the left and right hand sides.

Question 46

When will a limit tend towards infinity?

Answer 46

When a function is undefined at a certain value.

Question 47

What if the left and right hand limits are different?

Answer 47

If a function has two different limit values on the left and right hand sides, then it is discontinuous, and does not have an existing limit.

Question 48

What is the limit of a sum? ( lim (f(x)+g(x)) )

Answer 48

The limit of a sum equals the sum of the limits lim (f(x)) + lim(g(x))

Question 49

What is the limit of a difference? ( lim (f(x)-g(x)) )

Answer 49

The limit of a difference equals the difference of the limits lim (f(x)) - lim(g(x))

Question 50

What is a constant times a limit equal to? ( lim(cf(x)) )

Answer 50

It is equal to the limit times the constant ( clim(f(x)) )

Question 51

What is the limit of a product equal to? ( lim (f(x)*g(x)) )

Answer 51

The product of the limits (lim(f(x)))*(lim(g(x)))

Question 52

What is the limit of the quotient equal to? ( lim (f(x)/g(x)) )

Answer 52

The quotient of the limits, provided that lim(g(x)) is not equal to 0. lim((f(x)))/lim(g(x)))

Question 53

When would direct substitution of a limit be used?

Answer 53

When a function is continuous and the limit needs to be known exactly.

Question 54

How does squeeze theorem work?

Answer 54

Squeeze theorem works by putting a function g(x) between two easier functions f(x) and h(x). If f(x) and h(x) tend towards the same limit, then g(x) must do too.

Question 55

What is the symbol of, and what numbers does the real number group include?

Answer 55

They have the symbol R, and contain all numbers on the number line.

Question 56

What is the symbol of, and what numbers does the rational number group include?

Answer 56

They have the symbol Q, and contain all numbers that can be expressed as a fraction.

Question 57

What is the symbol of, and what numbers does the irrational number group include?

Answer 57

They have the symbol R\Q, and include only numbers that cannot be expressed as a fraction.

Question 58

What is the symbol of, and what numbers does the integer number group include?

Answer 58

They have the symbol Z, and include all numbers without decimal points. Essential all rational numbers where the bottom of the fraction is 1.

Question 59

What is the symbol of, and what numbers does the natural number group include?

Answer 59

They have the symbol N, and consist of all integers greater than 0.

Question 60

What is the sum to infinity when the answer is estimated to be less than 1?

Answer 60

x/(1-x)

Question 61

What is the sum to infinity when the answer is estimated to be greater than 1?

Answer 61

1/(1-x)

Question 62

What makes a decimal periodic? And what can be done as a result of this?

Answer 62

A decimal is periodic if, after a section of numbers, it has a section that repeats forever. This allows it to be expressed as a fraction, making it a rational number.

Question 63

What does “A only if B” actually mean?

Answer 63

A can only ever be true if B is true

Question 64

What does “A if B” actually mean?

Answer 64

Event A will occur if event B occurs, but event A can occur without B occurring. This is different to “only if”, in that an event A can occur without B occurring.

Question 65

What does “A if and only if B” mean?

Answer 65

A is true when B is true, and B is true when A is true.

Question 66

How is the first principle differentiation equation derived?

Answer 66

Understanding that a graph's gradient is equal to (f(x)-f(a))/(x-a) when x tends towards a, and substituting in x = a+h gives the equation f'(a)=lim(h to 0)(f(a+h)-f(a))/(h).

Question 67

When is a function differentiable?

Answer 67

If every value a in a set interval (c,d) has a correpsonding derivative value

Question 68

When is a function not differentiable?

Answer 68

The function will not be differentiable if at any point in the interval (c,d) it is discontinuous or has a vertical tangent line.

Question 69

What is an indeterminate form?

Answer 69

An indeterminate form is one in which a limit lim(x to a) (f(x)/g(x)) has both functions tending to 0 or infinity. This gives an answer of 0/0, of infinity/infinity which shows that the limit may exist but required further analysis.

Question 70

How do you solve an indeterminate form?

Answer 70

Using L'Hospital's Rule, as it states that the derivative of an equation will still tend towards the same limit. Therefore you can derive the top and bottom function seperately to find the limit. This is done as many times as necessary.

Question 71

What is an inequality?

Answer 71

An inequality is a mathematical statement in which real numbers are ordered using the symbols "<", ">", "≤", and "≥".

Question 72

What are the 9 rules of inequalities?

Answer 72

1. If x is a real number it must be greater than 0, equal to 0 or less than 0, and can only be one of these. 2. If y > x, then -y < -x 3. If x > 0 and y > 0 then xy > 0 4. If y > x and c is a real number, then y + C > x + C 5. If x > y and y > z then x > z 6. If x > 0 and u > v then xu > xv 7. If u>v>0, then u^2>v^2 8. If x > 0, then 1/x > 0 9. If x < 0 and y < 0, then xy > 0