FUNCTIONS AND SEQUENCES

What is a function?

• Math machine that uses an input to produce output numbers

• Numbers are put into a function (input) and numbers come out of the function (output)

What is the domain of the function?

• Set of inputs that a function can use is called the domain of the function

What is the range of the function?

• Set of all outputs that the function can generate is called the range of the function

in f(x) = y, what do variable x and y denote?

• x denotes the independent variable

• y denotes the dependent variable because it’s value depends on the value of x

For f(1)=3, f(2)=5, f(3)=7; what is the domain and range of the set?

• Set {1,2,3} is called the domain of f and set {3,5,7} is called the range of f

What is a compound function?

• A compound function is a function that is actually composed of two or more functions that are nesting in one another.

• For eg: f(f(x)), g(f(x)) etc.

For a function of the form f(x) = kxⁿ + c (n is +ve even integer and k is non zero), what is the range of f(x)?

• K > 0, range of f(x) is all real numbers >= c

• K < 0, range of f(x) is all real numbers <= c

How do you visually tell what is the domain and range of the graph?

• To find the domain of the graph, look at the graph from left to right

• To find the range of the graph, look at the graph from top to bottom

What is the vertical line test?

• The vertical line test can be used to determine if a graph represents a function.

• Inspect the graph to see if any vertical line drawn would intersect the curve more than once.

  • If any vertical line intersects the graph more than once, the graph does NOT represent a function.

  • If any vertical line intersects the graph only once, the graph does represent a function.

What is a sequence?

• A sequence is an ordered list of numbers in which the order of the numbers is explained by a formula

• Every sequence has a rule/formula that dictates how the sequence works

• In order to make any conclusion about the terms in the sequence, the rule must be known

• When we talk about sequences, we don’t use the function notation f(n). Instead, we use the sequence notation a(n)

What is an explicit formula?

• When the formula of the sequence is given in terms of n, it is called an explicit formula

• A_n = 3(n-1) or S_n = n² - 1

• If the formula of a sequence is given recursively, we must know ALL terms prior to it

What is an arithematic sequence?

• Sequence in which the difference between every pair of consecutive terms is the same

• An arithematic sequence has the formula a_n = a₁ + (n-1)d

  • a_n = nth term

  • a₁ = First term

  • d = common difference

What is a finite sequence and an infinite sequence?

• If a sequence has infinitely many terms, it is called an infinite sequence

• If a sequence has finite terms, it is called a finite sequence

How to calculate the sum of the first n terms of an arithematic sequence?

S_n = n/2 (a₁ + a_n)

What is a geometric sequence (or geometric progression)?

• A geometric sequence (or geometric progression) is one in which the ratio between every pair of two consecutive terms is the same

What is the formula of geometric series?

a_n = a₁ * r^n-1

• a_n = nth term

• a₁ = first term

• r^n-1 = common ratio

When the common ratio is not given, it can be obtained by dividing any term of the sequence (except first) by the term that precedes it

How do you calculate the maximum or minimum output of a function f(x) = ax² + bc + c (a ≠ 0)?

• If a > 0 , minimum output occurs when we input x = -b/2a

• If a < 0, maximum output of a function occurs when we input x = -b/2a

• When we determine the minimum or maximum value, we must plug this value back into the original function equation to determine the minimum or maximum output

For eg, what is the least possible value of f(x) where f(x) = 4x² - 8x + 3?

• Since a > 0, minimum value = -b/2a = 1

• Plugging this value back into the equation, we get -1 as the output which is the least possible value

How do you solve questions such as:
If f(x) = x², is it true that f(n+m) = f(n) + f(m) ?

• Let n = 2 and m = 1, f(2+1) = f(2) + f(1)

• Inputting these values back into the f(x) equation we get f(3) = 5

• Validating this by add x = 5 in f(x) equation, we get 9 as the output

• Since 5 ≠ 9, the statement is not true

How do you calculate unknown values on a number line that has equally spaced tick marks?

• When the tick marks are equally spaced, the difference between 2 consecutive tick marks is constant

• We must assign a variable to find the common difference between tick marks

• The difference can be labeled from left to right by adding the common difference, or from right to left by subtracting the common difference

How to deal with sequences with repeating patterns?

• List several terms at the beginning of the sequence to determine the nature of the pattern

• If the pattern repeats itself every r terms, the repeating pattern has a length of r

• You can similarly deal with questions asking you to find the sum or product of consecutive values in a sequence. Be mindful that many terms will cancel each other out to make calculation manageable