Name
Pre-Calculus Unit 1 Tentative Syllabus
Piecewise and Power Functions
DATE
Mon.,
Aug. 22
1 Day Procedures
Day 1: Sec 1.2 Definition of Function
Tues.,
Aug. 23
Day 2: Sec 1.4
Composition of Functions
WS 2 – Composition of Functions
Wed.,
Aug. 24
Day 3: Sec 1.2
Properties of Functions
WS – Properties of Functions
(not in packet)
Thurs.,
Aug. 25
Day 4: Sec 1.2
More Properties of Functions
WS 4 – Properties of Functions
Fri.,
Aug. 26
Day 5: Sec 1.3
Graph, Evaluate and Analyze Piecewise Functions
WS 5 – Piecewise Functions
Mon.,
Aug.29
Tues.,
Aug 30
Wed.,
Aug 31
Thurs.,
Sept. 1
Fri.,
Sept. 2
Mon.,
Sept. 5
TOPIC
st
QUIZ 1.1 – Functions and Composition
Day 6: Sec 1.3 More Piecewise Functions
Day 7: Sec 2.2 Graph and Analyze Power Functions
ASSIGNMENT
WS 1 – Functions
WS 6 – More Piecewise Functions
WS – Power Functions
(Legal size – not in packet)
QUIZ 1.2 – Graph and Analyze Piecewise Functions
Day 8: Sec 2.2 Model Power Functions
WS 8 – Model Power Functions
Day 9: Review
Finish packet.
TEST 1
NO SCHOOL – LABOR DAY HOLIDAY
Day 1 Homework
Definition of FUNCTION
1. A relation that assigns to each element x from a set of inputs, or
of outputs, or
, exactly one element y in a set
, is called a
.
2. Functions are commonly represented in four different ways,
,
, and
,
.
Determine whether the equation represents y as a function of x. Write yes or no.
3.
4.
5.
Is the relationship a function? Write yes or no.
8.
9.
6.
7.
Is the relationship a function? Write yes or no.
13.
14.
10.
11.
15.
Which sets of ordered pairs represents functions
from A to B? Explain.
16.
17.
Is the relationship a function? Explain your reasoning.
12.
Day 2 Notes/Homework Composition of Functions
The operation of combining functions (in the correct order) which has no counterpart in the algebra of real numbers is called
function composition. Notation for composition is ( f g )( x)  f ( g ( x)) , where the function g is applied first, then f.
GIVEN:
f ( x)
g ( x)
Use the graphs above to evaluate each expression and answer the questions that follow.
1) f (2)
2) g (0)
3) g (2)
4) g (2)
) f (0)
6) f (3)
7) f ( g (1))
8) g ( f (5))
9) f ( g (0))
10) f ( g (2))
11) f ( g (2))
12) f
13) g g  0 
14) g
g  1
f g  4
15) What is the Domain of f  x  ?
16) What is the Domain of g  x  ?
17) What is the Range of f  x  ?
18) What is the Range of g  x  ?
19) When is f ( g ( x))  8 ?
20) When is g
Use the table below to evaluate the following compositions.
x
-2
-1
0
f(x)
-5
-3
-1
g(x)
8
3
0

21) f g  1


22) g f  0


23) g f  3
1
1
-1

2
3
0
24) f
 f  2
f  x  4 ?
3
5
3
4
7
8

25) g g 1

Continue to next page…
Text p. 117, #11 – 27
In #11-#14, find
(f
g )(3) and ( g f )(2).
In #15-#22, find
( f ( g ( x)) and g ( f ( x)).
State the domain of each.
In #23-#30, find f(x) and g(x) so that the
function can be described as y = f(g(x)).
(There may be more than one decomposition.)
Day 4 Homework:
Properties of Functions
Given these basic functions, answer the following questions.
A. Identity Function
B. Quadratic Function
f ( x)  x 2
f ( x)  x


f ( x)  x 3
f ( x) 









































F. Absolute Value Function
G. Exponential Function
f ( x)  x
f ( x)  e























f ( x)  ln x



H. Logarithmic Function
x


1
x


f ( x)  x

D. Reciprocal Function

E. Square Root Function

C. Cubic Function


















Identify which of the 8 basic functions fit the given description:
1. have a vertical asymptote.
10. decreasing on interval (, 0) .
2. have a horizontal asymptote.
11. range is all Real numbers.
3. domain excludes zero.
12. has at least one local maximum.
4. domain consists of all nonnegative real numbers.
13. has at least one absolute minimum.
5. has at least one discontinuity.
14. is an even function.
6. is a continuous function.
15. is an odd function.
7. is bounded above.
16. as x  , f ( x) does not approach 
8. is bounded below.
17. as x  , f ( x)  0 .
9. increasing on the entire domain.
18. as x  , f ( x)  


Day 5 Notes:
GRAPHING PIECEWISE FUNCTIONS
1) Graph y   x  2  3 then highlight
2) Graph y  2  2 x then highlight
2
where the graph has a domain of x  1 .


3) On a new grid, graph only the
where the graph has a domain of x  1 .
highlighted parts of #1 and #2.
Is the new graph a function?
Explain.
































b) As x   , f  x  
?

This is the equation of that new
graph of #3.
Using your graph in #3, determine the following answers.
a) As x   , f  x  

?
2

 x  2   3 , x  1
f  x  

2  2 x , x  1
c) Is the graph of f  x  continuous? If not, explain why and state the type of discontinuity.
d) Does the graph of f  x  have a relative maximum or a relative minimum?
If it does, where does the relative maximum or relative minimum occur?
Graph the following piecewise functions. State the end behavior, whether the function is continuous or discontinuous, the type of
discontinuity (if it’s discontinuous), and any relative extrema.
1
 x3 , x  0
4) f  x    2
 x  2  1 , x  0

As x   , f  x  


As x   , f  x  







Continuous/Discontinuous


Type of discontinuity

Relative extrema

4  x 2 , x  1

5) f  x    3 , 1  x  2
 1

, x2
x2
As x   , f  x  


As x   , f  x  







Continuous/Discontinuous


Type of discontinuity

Relative extrema
Evaluate the piecewise function at each value without graphing.
 x  3 ,  3  x 1

6) f  x   2 , 1  x  5
a) f  0  =
 2
x 1 , x  5
d) f  7  =
b) f  3 =
c) f 1 =
e) f  5  =
f) f  5  =
7) Write an equation for the piecewise function.
Day 5 Homework
PIECEWISE FUNCTIONS
Graph the following piecewise functions. Determine whether it is continuous or not (and type of discontinuity), the domain and
range, any intercepts and the end behavior.
 x 2 if x  1
1) f ( x)  
3x  5 if 1  x  3


2 x  3 if x  1

3x  4 if 0  x  3
3) f ( x)  
4) f ( x)   x  1 if  1  x  2
-4 if  3  x  0
 2
if x  2
x
x0

  x,
2) f ( x)  

 x  3, x  1













































Continue to next page…

Evaluate without graphing.
4  x 2 , x  1

1, x  0
3
3
71) f ( x)  
8) f ( x )   x  , 1  x  3
2
 x , x  0
2
 x  3, x  3
Find f (1), f (0), f ( )
Find f (1), f (0), f (5)
Find f (1.5), f (1), f (3), f (4)
1
 , x0
6) f ( x)   x
 x, x  0
3  x, x  1
 2 x, x  1
5) f ( x)  
Find f (0), f (1), f (2.5)
Find f 1 , f 5 , f  4 , and f  0 for each of the graphs below. Write an equation for each piecewise function.
13)
14)
Day 6 Homework
15)
More PIECEWISE FUNCTIONS
Use the graphs of the piecewise functions h  t  and g  t  to answer the following questions.
1
7
The equation of the graph in h  t  on the interval (1,3] is y   t 2  t  1 .
2
2
1) Is h  t  continuous? If not,
where is it discontinuous?
4) On what interval(s) is
g  t  increasing?
2) What is the domain and
range of h  t  ?
5) On what interval(s) is
h  t  decreasing?
3) What is the domain and
range of g  t  ?
6) What is the absolute
maximum value of g  t  ?
Use the graphs of the piecewise functions h  t  and g  t  to answer the following questions.
1
7
The equation of the graph in h  t  on the interval (1,3] is y   t 2  t  1 .
2
2
7) Determine the value of each expression below.
a) h  3
b) g  3
c) g (1)
d) g  h  4  
e) h  g  3 
8) What is the absolute minimum value of h  t  ?
9) Solve the equation h  t   4 .
10) Write an equation for the graph of h  t 
11) Write an equation for the graph of h  t 
on the interval  7,1 .
on the interval 3,9
12) Write an equation, using three pieces,
for the graph of h  t  on its domain.
13) Using your own words, describe how you
would determine where h  t   0 ?
14) Using your own words, describe how you would
determine the interval(s) where g  t   2 ?
Day 8 Notes/Homework
Modelling Power Functions
Write the statement as a power function equation. Use k for the constant of variation if one is not given.
1. The area A of an equilateral triangle varies directly as the square of the length s of its sides.
2. The current I in an electrical circuit is inversely proportional to the resistance R, with constant of variation V.
3. Charles’s Law states the volume V of an enclosed ideal gas at a constant pressure varies directly as the absolute temperature T.
4. The volume V of a circular cylinder with fixed height is proportional to the square of its radius r.
Solve each problem.
5. BOYLE’S LAW: The volume of an enclosed gas (at a constant temperature) varies inversely as the pressure. If the pressure
of a 3.46-L sample of neon gas at a temperature of 302 K is 0.926 atm, what would the volume be at a pressure of
1.452 atm if the temperature does not change?
6. WINDMILL POWER: The power P (in watts) produced by a windmill is proportional to the cube of the wind speed v (in
mph). If a wind of 10 mph generates 15 watts of power, how much power is generated by winds of 20, 40, and 80 mph?