Chapter 20: Functions and Sequences

the numbers a functions uses are called ______ and the numbers it produces are called ________

input, output

domain

the set of all allowable inputs

if the domain of a function is not specified, then the domain will be the set of all real numbers except:

those numbers that produce a denominator of zero

those numbers that produce a negative radicand of an even-indexed root

range

the set of all possible outputs of the function

what is the range of f(x) √x +b

the set of all real numbers greater than or equal to b

what is the range of f(x) = x^n +c if n is a positive even integer

the set of all real numbers greater than or equal to c

what is the range of f(x) =x^n +c if n is a positive odd integer

all real numbers

what is the range of f(x) = ax+b if a cannot equal 0

all real numbers

what are the steps to determining the range of a function?

determine the function’s maximum possible output

determine the function’s minimum possible output

determine the values that cannot be outputs

when evaluating a compound function, work from the

inside out

compound function

composed of two or more nested functions

when a function is graphed, each point on the graph has coordinates

x, f(x)

the domain is the set of all possible ___ coordinates

x

the range is the set of all possible ___ coordinates

y

in many cases, the domain of a graphed function can be thought of as defining the ____ width of the graph, and the range can be though of as defining the graph’s ______ length

horizontal, vertical

when a question asks us to determine a function’s output when the input is an algebraic expression:

take the function and replace all values of x with the given algebraic expression and simplify

the process for solving symbolism problems is similar to evaluating functions for a specific value, in both cases, we

replace all instances of a certain variable with the given number

for all values of z, the symbol z^@ denotes the greatest integer less than or equal to z. what is the value of 5.1^^@ + (-3.752)^@ - 8^@

5.1 —> 5

-3.752—> -4

-8 = -8

5+(-4) + (-8) = -7

when symbolism questions require us to create and solve an equation…

create the required equation by replacing each instance of the variable with the given value

the key to correctly solving function word problems is

to carefully input the proper data into the function

sequence

an ordered list of numbers, and it is also a function in which the domain is all positive integers

what is the common notation for a sequence?

a_n

recursive formula

the nth term of a sequence a_n is given in terms of its previous term or terms

how can you “translate” this question: if sequence A is defined by a_n=a_n-1+a_n-2 for n≥3 and a₁=a₂=2 what is a_5?

a₁=a₂ —>the first two terms are both 1

a_n = a_n-1+ a_n-2 for n≥3 —> starting from the 3rd term, each term is the sum of the previous two terms

arithmetic sequence

a sequence where the difference between every pair of consecutive terms is the same

what is the equation to find an arithmetic sequence?

a_n = a₁ + (n-1)d

where a₁ is the first term and d is the common difference

what is the equation to find the some of the first n terms of an arithmetic sequence?

S_n = n/2 (a₁ + a_n)

geometric sequence

a sequence where the ratio between every pair of consecutive terms is the same

what is the equation for geometric sequences?

a_n= a₁ x r^n-1

where a₁ is the first term and r is the common ratio