Functions & Graphs (2.2)

Question 1

what is the modulus of a number |a|?

Answer 1

its non-negative numerical value

Question 2

what is the modulus function in general?

Answer 2

y = |f(x)|
when f(x) ≥ 0, |f(x)| = f(x)
when f(x) < 0, |f(x)| = -f(x)

Question 3

how do you sketch the graph of y = |f(x)| / y = |ax+b|?

Answer 3

sketch y = f(x) / y = ax+b
reflect the section of the graph below the x-axis (where f(x) < 0) in the x-axis
delete the parts below the x-axis

Question 4

how do you sketch the graph of y = f(|x|)?

Answer 4

sketch the graph of y = f(x) for x ≥ 0
reflect this in the y-axis

Question 5

how do you sketch the graph of f(x) = a|x+p| + q?

Answer 5

a describes the shape:
if a > 0, V shape
if a < 0, ^ shape
the vertex is (-p, q)

Question 6

describe how to find intersections of linear modulus graphs

Answer 6

2 non-parallel linear graphs intersect once only
if 1 or both of the graphs involves a modulus, 0, 1 or more than 1 intersections are possible
always sketch the graphs of the modulus functions to see the number of intersections

Question 7

define mapping & function

Answer 7

mapping: transforms one set of numbers into a different set of numbers

function: a mapping is a function if every input has a distinct output
either one-to-one or many-to-one (one-to-many mapping is not a function)

Question 8

define domain & range

Answer 8

domain: the set of all possible inputs for a mapping
restricting the domain can turn mapping into a function
range: the set of all possible outputs for the mapping

Question 9

describe composite functions

Answer 9

2 or more functions combined to make a new function
fg(x) means apply g first, then f
fg(x) = f(g(x))

Question 10

describe inverse functions

Answer 10

the inverse of a function performs the opposite operation to the original function - elements in the range of the original are converted back into the domain of the original

inverse functions only exist for one-to-one functions

ff-1(x) = f-1f(x) = x

the graphs of f(x) & f-1(x) are reflections of one another in the line y = x

the domain of f(x) is the range of f-1(x)
the range of f(x) is the domain of f-1(x)

Question 11

describe how to find the equation for the inverse function

Answer 11

change x’s to y’s & y’s to x’s
then rearrange to make y the subject

Question 12

combining transformations

Answer 12

apply stretch first - stretch inside brackets comes before stretch outside brackets

then apply translation - translation inside brackets comes before translation outside brackets

translation inside brackets comes before stretch outside brackets

Question 13

describe how to solve modulus equations

Answer 13

e.g. |ax + b|= k
ax + b = k & solve
ax + b > 0 –> check that the solution is in the domain for x
ax + b = -k & solve
ax + b < 0 –> check that the solution is in the domain for x

see Baldwin notes