Functions and Sequences Flashcards
what is the term for the set of all inputs for a function?t
the domain of the function
what is the term for the set of all possible outputs of a function?
the range of the function
what are the two major concerns when identifying the domain of a function on the GMAT?
1) cant take the square root of a negative number
2) cant divide by 0
how do you evaluate compound functions?
evaluate the innermost first then work your way out
how do you determine minimum and maximum values of a quadratic function?
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)
how to use the vertical line test to assess whether something is a function?
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
range of function questions
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.
symbolism in function evaluation
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
what is a good strategy for testing whether a certain function property holds for all input values?
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.
T/F: we need to be prepared to evaluate functions where the output is described in words
True
ex: g(x) is the largest prime number less than or equal to x. so g(8) = 7
what is a sequence?
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,…
every sequence has a rule or formula that governs the sequence
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
what is an arithmetic sequence?
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
how do you find the sum of the first n terms of an arithmetic sequence
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
commonly spaced tick marks on a line
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
see screenshot for how to handle more challenging tick mark examples
see screenshot
hard-ish problem with consecutively spaced tick marks
see screenshot
geometric sequence
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
what is the “length” of a repeating pattern?
if a pattern repeats itself every r terms, then its has a length of r
what are the two main types of questions involving repeating patterns?
1) find a specific terms in the sequence
2) find the sum of a set of consecutive terms in the sequence
question type: finding terms in a repeating sequence
solve enough terms to where you can identify pattern. use length of pattern to id right answer
question type: finding sums in repeating sequences
list out the first and last few terms in the range we are asked to add
cancelling terms in a sequence
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
