All SAT II Math II Resources
Example Questions
Example Question #1 :Solving Piecewise And Recusive Functions
Defineandas follows:
Evaluate.
by definition.
on the set, so
.
on the set, so
.
Example Question #1 :Solving Piecewise And Recusive Functions
Define functionas follows:
Give the range of.
The range of a piecewise function is the union of the ranges of the individual pieces, so we examine both of these pieces.
If, then. To find the range ofon the interval, we note:
The range of this portion ofis.
If, then. To find the range ofon the interval, we note:
The range of this portion ofis
The union of these two sets is, so this is the range ofover its entire domain.
Example Question #3 :Solving Piecewise And Recusive Functions
Define functionas follows:
Give the range of.
The range of a piecewise function is the union of the ranges of the individual pieces, so we examine both of these pieces.
If, then.
To find the range ofon the interval, we note:
The range ofonis.
If, then.
To find the range ofon the interval, we note:
The range ofonis.
The range ofon its entire domain is the union of these sets, or.
Example Question #4 :Solving Piecewise And Recusive Functions
Define functionsandas follows:
Evaluate.
Undefined
First, we evaluate. Since, the definition offoris used, and
Since
, then
Example Question #2 :Solving Piecewise And Recusive Functions
Define functionsandas follows:
Evaluate Evaluate.
Undefined
Undefined
First, evaluateusing the definition offor:
Therefore,
However,is not in the domain of.
Therefore,is an undefined quantity.
Example Question #6 :Solving Piecewise And Recusive Functions
Define functionsandas follows:
Evaluate.
Undefined
First, evaluateusing the definition offor:
Therefore,
Evaluateusing the definition offor:
Example Question #7 :Solving Piecewise And Recusive Functions
Define functionsandas follows:
Evaluate.
Undefined
First we evaluate. Since, we use the definition offor the values in the range:
Therefore,
Since, we use the definition offor the range:
Example Question #8 :Solving Piecewise And Recusive Functions
Define two functions as follows:
Evaluate.
By definition,
First, evaluate, using the definition offor nonnegative values of. Substitutingfor 5:
; evaluate this using the definition offor nonnegative values of:
12 is the correct value.
Example Question #9 :Solving Piecewise And Recusive Functions
Which of the following would be a valid alternative definition for the provided function?
None of these
The absolute value of an expressionis defined as follows:
for
for
Therefore,
if and only if
.
Solving this condition for:
Therefore,for.
Similarly,
for.
The correct response is therefore