Functions and Sequences

Question 1

what is the term for the set of all inputs for a function?t

Answer 1

the domain of the function

Question 2

what is the term for the set of all possible outputs of a function?

Answer 2

the range of the function

Question 3

what are the two major concerns when identifying the domain of a function on the GMAT?

Answer 3

1) cant take the square root of a negative number
2) cant divide by 0

Question 4

how do you evaluate compound functions?

Answer 4

evaluate the innermost first then work your way out

Question 5

how do you determine minimum and maximum values of a quadratic function?

Answer 5

for quadratic function in the form f(x) = ax^2+bx+c (where a != 0):
- if a > 0, the minimum of f occurs at x=-b/(2a)
- if a < 0, the maximum of of occurs at x=-b/(2a)

Question 6

how to use the vertical line test to assess whether something is a function?

Answer 6

a given x value can have only 1 corresponding y value if it has more than one it fails the vertical line test and is not a function

Question 6

range of function questions

Answer 6

generally in the form f(x)=k*x^n + c, where n is a positive, even integer, and k is nonzero.

The minimum value of x^n is zero (when x=0). Then, if k>0, then the range is all real numbers >=c, if k<0, the range is all real numbers <=c.

Question 7

symbolism in function evaluation

Answer 7

a#b = a+b-(a/b) -» pound symbol is to be taken as an operator/function describing how when “a operates on b” the following set of operations occurs to a and b

*see screenshot

Question 8

what is a good strategy for testing whether a certain function property holds for all input values?

Answer 8

plug in values for the given variables and see if the property holds true
ex: for f(x) = x^2 does f(m+n) = f(m) +f(n)? Let m =1, n=1, (1+1)^2 != 1^2 + 1^2 since 4 != 2.

Question 9

T/F: we need to be prepared to evaluate functions where the output is described in words

Answer 9

True
ex: g(x) is the largest prime number less than or equal to x. so g(8) = 7

Question 10

what is a sequence?

Answer 10

an ordered list of numbers in which the order of the numbers is explained by a formula (a sequence is actually a function whose domain is all positive integers)

ex: sequence f(n) 3*(n-1) for n =1,2,3,…. yields, 0,3,6,9,…

Question 11

every sequence has a rule or formula that governs the sequence

Answer 11

without the rule we CANNOT make any conclusions about the value of any term in the sequence. if first few terms are given, but the rule is not, we cannot infer any further terms

Question 12

what is an arithmetic sequence?

Answer 12

sequence in which the difference between every pair of two consecutive terms is the same. it has formula:

a_n = a_1 + (n-1)*d, where a_n is the Nth term in the sequence, a_1 is the first term of the sequence, and d is the common difference

Question 13

how do you find the sum of the first n terms of an arithmetic sequence

Answer 13

sum of the first n terms of an arithmetic sequence is:

S_n = n* ((a_1+a_n)/2) – note this is leveraging the fact that for an evenly spaced set of numbers the average can be calculated as (first+last)/2, then multiply that average by the number of items in the sequence to get the sum

Question 14

commonly spaced tick marks on a line

Answer 14

if x and y are consecutive values on a number line with equally spaced tick marks and x<y, then the common difference between tick marks is y-x

Question 15

see screenshot for how to handle more challenging tick mark examples

Answer 15

see screenshot

Question 16

hard-ish problem with consecutively spaced tick marks

Answer 16

see screenshot

Question 17

geometric sequence

Answer 17

one in quich the ratio between every pair of consecutive terms is the same (this ratio is called the common ratio) - if this common ratio is not given it can be obtained by dividing any term in the sequence by the term that directly precedes it

formula: a_n = a_1 * r^(n-1), where a_n is the Nth term, and a_1 is the first term, and r is the common ratio

Question 18

what is the “length” of a repeating pattern?

Answer 18

if a pattern repeats itself every r terms, then its has a length of r

Question 19

what are the two main types of questions involving repeating patterns?

Answer 19

1) find a specific terms in the sequence
2) find the sum of a set of consecutive terms in the sequence

Question 20

question type: finding terms in a repeating sequence

Answer 20

solve enough terms to where you can identify pattern. use length of pattern to id right answer

Question 21

question type: finding sums in repeating sequences

Answer 21

list out the first and last few terms in the range we are asked to add

Question 22

cancelling terms in a sequence

Answer 22

when asked to find the sum or product of several terms in a sequence, keep in mind that in many cases, most of the terms will cancel each other to make the calculation manageable

*when calculating terms needed for a sum in which you suspect values might cancel, do not immediately simplify/evaluate subtraction or additions as this may obscure what can be cancelled