16) Functions and Sequences

Question 1

Functions

Answer 1

Input some number for x and output some number for f(x)

A function can only have ONE OUTPUT, y, for each unique input —> vertical line test to determine if a graph is a function

Read as “f of x” NOT times

Terms:
> Domain = set of ALL inputs into a function —> Pay attention to RESTRICTIONS BEFORE you simplify functions (cannot divide by 0, cannot square root a negative value)
> Range = set of ALL outputs of a function –> pay attention to quadratic functions that have minimum or maximum y values
> inputs –> independent variable (x)
> outputs –> dependent variable –> f(x)

f(3) —> what is the value of the function when the input is 3?
f(x+1) –> what is the value of the function when the input is x+1?
f(anything) –> plug in anything wherever x is

Question 2

“all real numbers” meaning for domain of a function?

Answer 2

Numbers are either Real numbers or Imaginary numbers

Real numbers include:
> rational numbers
> irrational numbers like sqrt(2)

Imaginary numbers are like taking the square root of a negative

Question 3

Composite (compound) functions - what are they?

Answer 3

Functions nested inside another function

e.g., f(g(x)) –> output of one function becomes the input of another function
e.g., f(f(x)) (iterated function operating on one function only)

To solve, work from INSIDE OUT

Question 4

If f(x) = x + x^2/3x + 4, which of the following numbers cannot be in the domain of f(f(x))?

I. 3
II. 0
III. -3

Answer 4

Domain –> cannot divide by 0 and take square root of a negative value

Looking for invalid numbers for INNER FUNCTION and OUTER FUNCTION

II and III

0 makes inner function invalid
While -3 makes inner function equal 0, so makes outer function invalid

Question 5

Determining the range of a function

Answer 5

Likely function will be in the form f(x) = ax^n + k, where n is a positive even integer and a is nonzero

Because the minimum value of x^n is 0, look to the SIGN OF k to determine the range of the function:

If a > 0, range is ALL real numbers >= k (opens UP like a U)
If a < 0, range is All real numbers <= k (opens DOWN)

Question 6

Vertical line test

Answer 6

Useful when given a graph and asked to determine which graph depicts a FUNCTION

Concept: A function can ONLY have ONE OUTPUT, y, for EACH unique input (one input, one output)
> If an x value (input) is in the domain of the function, that x-value can only produce ONE Y-value (output)
> If a graph IS indeed the graph of a function, then any vertical line drawn can only intersect the graph at exactly one point or at no points
> If a graph fails the vertical line test, then the graph cannot be the graph of a function (e.g., circle, a vertical line)

Question 7

Symbolism in functions

e.g., is @x = x^2 + 1 and y =/ 3, what is the value of y such that @y = @3?

Answer 7

Can have functions containing:
> more than one variable (e.g., f(x, y) = x^2 + y^3)
> using symbols (e.g., x@y = x^2 + y^3)

In symbolism problems, make sure you correctly follow the RULE the operator provides
> replace variables with values and perform operation

e.g., @3 = 3^2 + 1 = 10
@y = y^2 + 1

@y = @3 —> 10 = y^2 + 1
9 = y^2
y = +/- 3

since y =/ 3, y = -3

Question 8

Functions in word problems

Answer 8

Functions in word problems tend to model some useful scenario

e.g., population growth modelled via P(t) = 100*2^t
e.g., height of a ball thrown vertically upwards modelled via h(t) = -16t^2 + vt + h

Strategy for solving: carefully input the proper data into the function

Question 9

When an object is thrown directly upward from an initial height of h0 feet with an initial velocity of v0 ft/sec, its height above the ground as a function of time t in seconds is given by h(t) = -16t^2 + v0t + h0. A 6-ft man throws a baseball directly upward from an initial height of 4 ft with an initial velocity of 18 ft/sec. After 1 second, the height of the ball above the ground will be what?

Answer 9

Word problems involving functions:

v0 = 18
t = 1
h0 ** = “6-ft man throws a baseball directly upward from an initial height of 4 ft” = h0 is 4 (not 6+4)**
> man throws baseball from an initial height of 4ft

Question 10

What are sequences?

Answer 10

An ORDERED list of numbers explained by a formula
> a sequence is actually a FUNCTION in which the domain is ALL positive integers

Special sequence notation, an, where n = 1 is the first term

Caveats:
> it is very important to understand that every sequence has a RULE or formula that governs the sequence. WITHOUT THIS RULE we cannot make any conclusions about the value of any term in the sequence

e.g., if only the first few terms of a sequence are given, but the rule is NOT given, we CANNOT determine any further the terms in the sequence

TIPS:
> when stuck, JUST WRITE OUT terms in the sequence

Question 11

If x is the fifth term of some sequence an, what is the value of x?

(1) the first four terms of an are 2, 4, 16, 256
(2) each term of an from the 2nd term on is obtained by squaring the term preceding it

Answer 11

Sequences –> NEED to know the RULE in order to predict (don’t assume the rule)

an: _ _ _ _ x

x=?

(1) Looks like factor is squaring BUT CANNOT determine for sure

NS

(2) need to know term preceding x

(3) Sufficient

Question 12

What is a recursive notation

Answer 12

A formula of a sequence in which the nth term is based on the PREVIOUS term or terms (An-1 notation instead of n)
> could be arithmetic or geometric sequence
> NEED to know the FIRST TERM or PRIOR TERM of the sequence (otherwise, cannot determine the terms)
> unlike “explicit form”, we can determine any term with any n

e.g., a1 = 2 and an = 3*an-1, for n>=2

so terms in the sequence are:
a1 = 2
a2 = 32 = 6
a3 = 36 = 18
a4 = 3*18 = 54

Question 13

Given the formula for a sequence an = 2n+3 for n>=1, which of the following formulas is an equivalent formula?

(1) A1 = 3 and An = An-1 + 2 (n>=2)
(2) B1 = 2 and Bn = Bn-1 + 3 (n>=2)
(3) C1 = 5 and Cn = Cn-1 + 2 (n>=2)
(4) D1 = 5 and Dn = Dn-1 + 3 (n>=2)
(5) E1 = 5 and En = 2En-1 + 3 (n>=2)

Answer 13

Recursive notation and sequences
> formula is given in “explicit form”
> answer choices are given in “recursive form”

Strategy to solve:
> Using original sequence formula, list out first 5 terms:

a1 = 5
a2 = 7
a3 = 9
a4 = 11
a5 = 13

Each term increases by +2

Then find answer choice that creates the SAME sequence of terms (start at 5, increases by 2)

Ans C
C1 = 5
C2 = C1 + 2 = 7
C3 = C2 + 2 = 9
etc.

NOT A, because A1 = 3 (not 5):
A1 = 3
A2 = A1 + 2 = 5
A3 = A2 + 2 = 7

Question 14

Arithmetic sequence

Answer 14

Sequence where the DIFFERENCE between every pair of CONSECUTIVE terms is the same

Any term: an = a1 + (n-1)*d

FOR ANY STARTING TERM: an = ak + (n-k)*d

(For linear growth Qs, represented as Ht = H0 + t*d, where t >= 1)

Sum (arithmetic SERIES): average * N terms
= (a1 + an)/2 * N

Question 15

Geometric sequence

Answer 15

Sequence where the RATIO between every pair of CONSECUTIVE terms is the same

Common ratio

an = a1*r^(n-1)

Geometric Series: [a1*(1-r^n)]/(1-r)

Question 16

The common difference in an arithmetic sequence Sn is 3. If the first term of the sequence is 4, what is 2*(S5 - S4) + 1?

Answer 16

Arithmetic sequence

d = 3

SUBTRACTION between any two terms = RANGE of those terms = (N - 1)*d

S5 - S4 = (2 - 1)*3 = 3

2*3 + 1 = 7

Question 17

Arithmetic sequences –> dealing with subscripts

e.g., initial height is h, what is the height after t minutes

e.g., sequence is x0, x1, x2 .. xk…xn. What is the expression for xk and xn?

Answer 17

POSITION start at 1

FIRST VALUE = Position 1
Last value = Position N

Last term = A1 + (n-1)*d

THIS FORMULA: An = Ak + (n-k)*d
—–> n-k = last term - first term = number of intervals between them

e.g., initial height is h, what is the height after t minutes
h, h1, h2 … ht

Therefore ht = h + (t)*d

e.g., sequence is x0, x1, x2 .. xk…xn. What is the expression for xk and xn?

Therefore xk = x0 + (k)d
xn = x0 + nd
xn = xk + (n-k)d
xn = x1 + (n-1)d
xn = x2 + (n-2)*d

Question 18

a, b, c, d (where a < b < c < d) are four consecutive terms of an arithmetic sequence and if c is removed, then a, b, d are three consecutive terms of a geometric sequence. If a = 2, what is the value of c?

Answer 18

Arithmetic and geometric SEQUENCE

Strategy for solving:
> create EQUATIONS by setting terms EQUAL TO EACH OTHER (need at least 2 equations to solve for the constant difference and the constant ratio)

Sequence A: 2, b, c, d
Sequence B: 2, b, d

b = b
2 + x = 2*r

d = d
2 + x3 = 2r^2

SOLVE: create a quadratic equation
0 = (r-1)*(r-2)

so r = 1 or 2

therefore x = 2 or x = 0 (not possible because a < b < c < d)

Therefore x = 2, and c = 2 + x*2 = 6

Question 19

If f(a+b) = 2a^2 + 2b^2 + 4ab for all real numbers a and b, what is f(5)?

Answer 19

Functions where argument is an expression
> a+b = 5 (COMBO of variables)
> we cannot assume b = 0 and a = 5, or b = 5 and a = 0
> we want to REPLACE a+b with 5 somehow

Notice how the function can be represented as:
f(a+b) = 2(a+b)^2

Now we can replace a+b with 5

f(5) = 2*5^2 = 50