Transcript Standard III -- Apply concepts related to functions

Standard III -- Apply concepts
related to functions
First some general
notes on functions.
Background notes on Functions:
 Relation – a set of ordered pairs.
 Domain – the first number in an ordered pair.
 Range – the second number in an ordered pair.
 When stating domain and range, never repeat a number.
Example {(2,3), (2,4), (7,4), (5,8), (3, 9)}
 Domain = {2,3,5,7}
 Range = {3,4,8,9}
 Notice the numbers are listed from least to greatest.
 Domain is the first number in an ordered pair and is
considered the “x”.
 Range is the second number in an ordered pair and is
considered the “y”.
 Think alphabetical order: Domain and Range; x and y
 D before R and x before y
How to tell if a relation
is a function. . .
 Remember the definition of Domain or “x” . . . When
determining if a relation (a set of ordered pairs) is a
function, it all depends on the answer to this question:
 Does a number a repeat in the domain or “x” position?
 If you say yes a number repeats, then the relation is
NOT a function. If you say no a number does not
repeat, then the relation IS a function.
 {(1,2), (2,3), (4,5)} Domain is {1,2,4} did you repeat a
number – NO, then it IS a function.
Find the range given domain?
 When a question has the phrase, find the range given
the domain it is a simply PII problem (Plug It In).
 Example: Find the range of y = 2x + 4 if domain is
{1,2,4}.
 Since there are three numbers given in the domain,
that means there are three answers. Write the
equation three times.
 Then PII the domain… see next slide.
Solve y = 2x + 4
 y = 2 x + 4 domain is {1,2,4}
 y=2(1)+4 »
y=2+4
 y=2(2)+4 » y=4+4
 y =2 ( 4 ) + 4 » y = 8 + 4
» y=6
» y =8
» y = 12
 Y remember represents range so you are basically
solving for “y”.
 The final answer is y or range is {6,8, 12}
A Quick way to determine if a
graph is a function…
 If you have a U or a V that opens up or down, then the
graph IS a function.
 If you have a U or a V that opens sideways, then it is
NOT a function.
 Horizontal Lines, then it IS a function.
 Vertical Lines, then it is NOT a function.
All other shapes?????
 Other shapes on graphs can be determined if it is or is
not a function by using the vertical line test.
 A vertical line is drawn on the shape. The vertical line
can only touch or cut the shape once and only once to
be a function.

Not a function

IS a function
Mappings?????
 A mapping is usually two vertical boxes or two ovals
with numbers inside of each. This is another method
to list a relation (set of ordered pairs).
 The first box or oval represents the x or domain.
 The second box or oval represents the y or range.
 Now remember your definition of a function – x or the
first number cannot repeat…so only ONE LINE can
come from each number in the X column. Meaning
each number in the X column can be matched with
ONLY one number in the Y column.
Standard 3 – Objective 1
Answer is A.
 Remember that a graph with a U or a V opening up, is
ALWAYS a function.
 Choice B is a U opening sideways which is NEVER a
function.
 Choice C and D are an oval and circle which are
NEVER a function.
2. Which of these mappings
is not a function?
Answer: C
 Look at column X, notice that the number 2 is mapped
to a 2 and a 4 in column f(x).
 F(x) is another way to write y.
Which of these represents the
data in the table?
Answer is B.
 This is a PII problem (plug it in).
 Start with equation A. Plug in the first x value into the
equation and perform Order of Operation. The
answer should match the first y value.
 Remember for an equation to be the correct one, each
of the x and y values must check for the equation.
Answer is D
 This is a PII problem.
 Notice when you plug in 0 into the x and solve the
problem the y or the answer is -5. That means the
answer is either C or D.
 After checking all the values in C and D, choice D is
the only choice which all three x values matches all
three y values.
Answer is A.
 The best way to solve this type of problem is to write
the mapping as ordered pairs.
 (0,1), (1,2), (2,3), (3,4)
 Now plug the x value into the equation and solve for y.
 The only equation that checks for every ordered pair is
A.
Answer is D.
 This is a PII problem as well. Plug in the first x value
into the x into the equation.
 Remember to take the absolute value first before
adding 1.
 The answer you find should match the first y value.
 Remember every value MUST check.
Answer is C.
 Choice C is the only relation that the X value does
NOT repeat in the x position.
Answer is B.
 Choice B is a V that opens down.
Answer is B.
 Remember a vertical line is NEVER a function.
Standard 3 Objective 2
Answer is C.
 Range is the second number in an ordered pair.
 Underline the second number in each ordered pair and
write the number on your paper. Remember do not
write a number more than ONCE.
 Answers are listed from least to greatest.
Answer is D.
 This is a plug it in problem.
 Plug in the first value for X and then apply order of
operations to the problem.
 y = 3x² - 5
{-2,0,1}
 y = 3(-2)² - 5
 y = 3(4) – 5
 y = 12 – 5
 y=7
NOTICE ONLY CHOICE B
HAS A 7 IN THE ANSWER.
Answer is A not D.
 If you calculated answer D then you have a common
mistake.
 Look closely at the problem. Did you notice there is a
negative sign in front of the x²? This is really -1 x².
Remember order of operations you must do the
exponents before multiplying. Therefore, you first
must square the number then multiply by -1.
 Look at the next slide for the step by step procedure for
solving this problem.
f(x) = -x² + 2x – 3 , what is f(4)
 The f(4) is the plug in value so you will plug in 4 for x.
 f(x) = -x² + 2x – 3





=
=
=
=
=
-1(4)² + 2(4) – 3
-1 (16) + 8 – 3
-16 + 8 – 3
-8 – 3
-11
Answer is D.
 Notice the problem is stating RANGE which is the
y-axis not the x-axis which most students mistakenly
read.
 Draw a line from each dot to the y-axis and read the
number on the y-axis from least to greatest.
 Also notice the dots are solid, therefore the symbols for
less than should have a line under each one.