Functions & Graphs (2.2) Flashcards
(20 cards)
what is the modulus of a number |a|?
its non-negative numerical value
what is the modulus function in general?
y = |f(x)|
when f(x) ≥ 0, |f(x)| = f(x)
when f(x) < 0, |f(x)| = -f(x)
how do you sketch the graph of y = |f(x)| / y = |ax+b|?
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
how do you sketch the graph of y = f(|x|)?
sketch the graph of y = f(x) for x ≥ 0
reflect this in the y-axis
how do you sketch the graph of f(x) = a|bx+p| + q?
a describes the shape:
if a > 0, V shape
if a < 0, ^ shape
the vertex is (-p/b, q)
describe how to find intersections of linear modulus graphs
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
define mapping & function
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)
define domain & range
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
describe composite functions
2 or more functions combined to make a new function
fg(x) means apply g first, then f
fg(x) = f(g(x))
describe inverse functions
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)
GIVE DOMAIN WITH THE INVERSE FUNCTION
describe how to find the equation for the inverse function
change x’s to y’s & y’s to x’s
then rearrange to make y the subject
GIVE THE DOMAIN
combining transformations
inside brackets, apply translations then stretch
outside brackets, apply stretch then translations
Csin(Bx+A) + D
A—>B—>C—>D
describe how to solve modulus equations
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
can only find inverse if the original function is one-to-one
can only find inverse graphically if the new mapping is a function
for quadratic functions, complete the square before finding the inverse
when is a piecewise mapping not a function?
when the domain for one part of the mapping is e.g. x>4 & the other is e.g. x<4
for g(x), there is no output for x=4 so not a function
not continuous
how to find range of piecewise function
sketch
when doing composite functions with piecewise functions: if the output of the 1st function is in the other equation of the piecewise function, use that part
define function
each value of x maps exactly to 1 value of y
reciprocal graphs are not functions unless the domain is limited
must exclude x =
at asymptotes to make it a function
