Functions and Sequences

Question 1

Function Definition

Answer 1

A function from a set X to a set Y assigns to each element of X exactly one element of Y

Question 2

Domain

Answer 2

Definition: Set of all inputs that a function can use (THINK: domain = x-value → XD)

Domain example: f(x) = x³ / x² - √x
→ Domain is x > 0 (all positive real numbers)

Domain tricks:

1. Can’t take the square root of a negative number
2. Can’t divide by 0

Question 3

Range

Answer 3

Set of all outputs that the function can generate

For a function of the form f(x) = kxⁿ + c where n is a positive even integer and k is nonzero:

• If k > 0 → range = all real numbers ≥ c
• If k < 0 → range = all real numbers ≤ c

Example: What’s the range of f(x) = 4 - 5x⁶? → All real numbers ≤ 4

EASIER WAY to conceptualize: Solve the equation for x = ______

Question 4

Vertical Line Test

Answer 4

• If a graph is the graph of a function, then any vertical line drawn can only intersect the graph at exactly one point or at no points.
• The x-value (input) can produce only one y-value (output)
• A function can only have one output, y, for each unique input, x.

Question 5

Sequence

Answer 5

• A sequence is an ordered list of numbers in which the order of the numbers is explained by a formula.
• A sequences is actually a function
• e.g., 0, 3, 6, 9, 12 → a_n = 3(n - 1)
• If the first few terms of a sequence are given, but the rule is not given, we cannot determine any further terms in the sequence.

Question 6

Sequence: Recursive Notation

Answer 6

• Recursive: e.g., a_n = 3a_n-1
  
  • To find a term, we must know ALL the terms prior to it
• Explicit: a_n = n + 1 (not recursive)
  
  • To find a term, we don’t need to know the terms prior to it.

Question 7

Arithmetic Sequence

Answer 7

Difference between every pair of two consecutive terms is the same

a_n = a₁ + (n - 1)d

Sum: S_n = (a₁ + a_n)n / 2

Question 8

Geometric Sequence

Answer 8

Ratio between every pair of two consecutive terms is the same

a_n = a₁ x r^n-1