M – Functions, Lesson 2, Function Notation, Evaluating Functions (r. 2018)
FUNCTIONS
Function Notation, Evaluating Functions
Common Core Standard
F-IF.2 Use function notation, evaluate functions for
inputs in their domains, and interpret statements that
use function notation in terms of a context.
Next Generation Standard
AI-F.IF.2 Use function notation, evaluate functions for
inputs in their domains, and interpret statements that use
function notation in terms of a context.
LEARNING OBJECTIVES
Students will be able to:
1) use function notation,
2) evaluate functions for specific input values, and
3) use function notation in context.
Overview of Lesson
Teacher Centered Introduction
Overview of Lesson
- activate students’ prior knowledge
- vocabulary
- learning objective(s)
- big ideas: direct instruction
- modeling
Student Centered Activities
guided practice
Teacher: anticipates, monitors, selects, sequences, and
connects student work
- developing essential skills
- Regents exam questions
- formative assessment assignment (exit slip, explain the math, or journal
entry)
VOCABULARY
function notation
dependent variable
independent variable
composition of functions
BIG IDEAS
Function Notation
In function notation, f (x) is used instead of the letter y to denote the dependent variable. It is read as “f of
x” or “the value f(x) is a function of x,” which is the independent variable. Other letters may also be used.
There are four primary advantages to using function notation:
1) The use of function notation indicates that the relationship is a function.
2) The use of function notation explicitly defines which variable is the dependent variable and which
variable is the independent variable.
3) The use of function notation simplifies evaluation of the dependent variable for specific values of
the independent variable.
Example: If
4) The use of function notation allows greater flexibility and specificity in naming variables.
Example #1: If total cost is a function of the number of pencils bought, a function rule might
begin with C(p)=.
Example #2: If miles driven at a constant speed is a function of hours driving, a function rule
might begin with M(h)=.
When graphing using function notation, the label of the y-axis is changed to reflect the function notation
being used.
Evaluating Functions
To evaluate a function for a specific input, simply replace the dependent variable with the desired input
throughout the function.
Example: Given the function , find the value of as follows:
Composition of Functions
Some functions are defined using other functions. Such functions are called compositions of
functions. For example, if
(
)
2fx x=
and
( ) ( )
3gx fx=
, then the function
( )
gx
is defined in
terms of the function
( )
fx
. Since we know that
( )
2fx x=
, we can use substitution to write
( ) ( )
32gx x=
.
DEVELOPING ESSENTIAL SKILLS
Evaluate the following functions for the given input values:
( )
( )
( )
( )
(
)
( )
23
1
2
3
4
5
fx x
f
f
f
f
f
= +
=
=
=
=
=
( )
( )
( )
( )
( )
(
)
31
1
2
3
4
5
fx x
f
f
f
f
f
= −
=
=
=
=
=
(
)
( )
( )
( )
( )
( )
2
23
1
2
3
4
5
fx x x
f
f
f
f
f
=++
=
=
=
=
=
( )
( ) ( )
( )
( )
( )
( )
( )
2
23
1
2
3
4
5
fx x
gx fx
g
g
g
g
g
= +
=
=
=
=
=
=
ANSWERS
( )
( )
( )
( )
( )
( )
23
15
27
39
4 11
5 13
fx x
f
f
f
f
f
= +
=
=
=
=
=
( )
( )
(
)
(
)
( )
( )
31
12
25
38
4 11
5 14
fx x
f
f
f
f
f
= −
=
=
=
=
=
( )
( )
(
)
(
)
( )
( )
2
23
16
2 11
3 18
4 27
5 28
fx x x
f
f
f
f
f
=++
=
=
=
=
=
( )
( ) ( )
( )
( )
( )
( )
( )
2
23
1 25
2 49
3 81
4 121
5 169
fx x
gx fx
g
g
g
g
g
= +
=
=
=
=
=
=
REGENTS EXAM QUESTIONS (through June 2018)
F.IF.A.2: Function Notation, Evaluating Functions
408) Given that , find if .
409) The graph of is shown below.
Which point could be used to find ?
1)
A
3)
C
2)
B
4)
D
410) The value in dollars, , of a certain car after x years is represented by the equation
. To the nearest dollar, how much more is the car worth after 2 years than after 3 years?
1)
2589
3)
15,901
2)
6510
4)
18,490
411) If , which statement is true?
1)
3)
2)
4)
412) The equation to determine the weekly earnings of an employee at The Hamburger Shack is given by ,
where x is the number of hours worked.
Determine the difference in salary, in dollars, for an employee who works 52 hours versus one who works
38 hours. Determine the number of hours an employee must work in order to earn $445. Explain how
you arrived at this answer.
413) If , then
1)
1
3)
2)
-2
4)
414) Lynn, Jude, and Anne were given the function , and they were asked to find . Lynn's
answer was 14, Jude's answer was 4, and Anne's answer was ±4. Who is correct?
1)
Lynn, only
3)
Anne, only
2)
Jude, only
4)
Both Lynn and Jude
415) If , what is the value of ?
1)
11
3)
27
2)
17
4)
33
416) For a recently released movie, the function models the revenue earned, y, in millions of
dollars each week, x, for several weeks after its release. Based on the equation, how much more money,
in millions of dollars, was earned in revenue for week 3 than for week 5?
1)
37.27
3)
17.06
2)
27.16
4)
10.11
417) If , then is
1)
315
3)
159
2)
307
4)
153
SOLUTIONS
408) ANS:
Step 1. Understand this as a composition of functions problem.
Step 2. Strategy: Substitute the expression for f(x) into the equation for g(x).
Step 3. Execution of Strategy.
and
PTS: 2 NAT: F.IF.A.2 TOP: Functional Notation Evaluating Functions
409) ANS: 1
Strategy: Understand that the meaning of is the value of y when , then eliminate wrong
answers.
Choose answer choice A because represents with coordinates . .
Answer choice b is wrong because if represents .
Answer choice c is wrong because if represents .
Answer choice d is wrong because if represents .
PTS: 2 NAT: F.IF.A.2 TOP: Functional Notation Evaluating Functions
410) ANS: 1
Strategy #1
Input into a graphing calculator and press enter.
Strategy #2: Input the function rule in a graphing calculator and obtain the value of the car after 2 years
and 3 years from the table of values. Then, compute the difference.
STEP 1: Input the function rule and obtain data from the table of values.
STEP 2: Compare the value of the car after 2 years and after 3 years.
The car is worth $18,490 after 2 years.
The car is worth $15,901 after 3 years.
The difference is
PTS: 2 NAT: F.IF.A.2 TOP: Functional Notation Evaluating Functions
411) ANS: 2
Strategy #1: Input into a graphing calculator and inspect the table of values.
x
f(x)
3
13
-2
3
-15
211
Strategy #2: Manually calculate the answer.
PTS: 2 NAT: F.IF.A.2 TOP: Functional Notation Evaluating Functions
412) ANS:
a) The difference in salary, in dollars, for an employee who works 52 hours versus one who works 38
hours, is $200.
b) An employee must work 43 hours in order to earn $445. See work below.
Strategy: Part a: Use the piecewise function to first determine the salaries of 1) an employee who works
52 hours, and 2) an employee who works 38 hours. Then, find the difference of the two salaries.
Working 38 Hours
Working 52 Hours
The difference between the values of and is $200.
Strategy: Part b: The employee must work more than 40 hours, and compensation for hours worked in
excess of 40 hours is found in the second formula and is equal to $15 per hour. The compensation worked
in excess of 40 hours is , so
The employee must work a total of 43 hours. The employee receives $400 for the first 40 hours and $45
for the 3 hours in excess of 40 hours.
PTS: 4 NAT: F.IF.A.2 TOP: Functional Notation Evaluating Functions
413) ANS: 3
Strategy: Substitute for x, and solve.
PTS: 2 NAT: F.IF.A.2 TOP: Functional Notation Evaluating Functions
414) ANS: 1
PTS: 2 NAT: F.IF.A.2 TOP: Functional Notation
415) ANS: 3
PTS: 2 NAT: F.IF.A.2 TOP: Functional Notation
416) ANS: 3
Strategy #1. Input the function rule in a graphing calculator, then use the table of values to identify the
revenues earned in weeks 3 and 5, then compute the difference.
The table of values shows that the movie earned 27.163 million dollars in week 3.
The table of values shows that the movie earned 10.107 million dollars in week 5.
The difference is
Strategy #2. Use a graphing calculator to evaluate the expression , which
equals 17.056..
PTS: 2 NAT: F.IF.A.2 TOP: Functional Notation Evaluating Functions
417) ANS: 4
Strategy: Substitute and solve.
Notes
Left
Expression
Sign
Right Expression
Given
=
Substitute 9 for x
=
Exponents and Radicals
=
Simplify
=
162-9
Simplify
=
153
PTS: 2 NAT: F.IF.A.2 TOP: Functional Notation
