Honors Precalculus
1
Name
Unit 1: Functions and Their Inverses
Date: _____________________
Section 2.1: Evaluating Functions
Find the following for each function:
a.
f  0
b.
f 1
g. f  2 x 
1)
f  x    2x2  x  1
c.
f  1
h.
d.
f x
e.  f  x 
f  x  h
2) f  x  
x
x 1
2
f.
f  x  1
Honors Precalculus
2
Name
Unit 1: Functions and Their Inverses
Date: _____________________
Warm-Up 1
Evaluate the following function.
1. f ( x)  2 x  3x  4
2
a) f (4)
b) f ( x )  f (4)
d)  f ( x )
e) f (2 x )
f) f ( x 1)
g) f (x + h)
h) f (x + h)- f (x)
i)
c) f (  x )
f (x + h) - f (x)
h
State the domain of the following functions in interval notation.
2. f ( x)   x  2

3. f ( x)  x 2  3x  28

1
Honors Precalculus
3
Name
Unit 1: Functions and Their Inverses
Date: _____________________
Section 2.1: Difference Quotient
Find the difference quotient using the following functions.
1) f  x    7 x  31
2) f  x    2 x  x  1 ****
2
3) f  x  
2x2  1
4x  5
Honors Precalculus
Unit 1: Functions and Their Inverses
4
Name
Date: _____________________
Honors Precalculus
Unit 1: Functions and Their Inverses
Section 2.1: Functions
5
Name
Date: _____________________
Honors Precalculus
6
Name
Unit 1: Functions and Their Inverses
Date: _____________________
Determine whether the relation is a function. If they are, give the domain and range.
Use appropriate notation.
1) Employees of Sara’s Pre-Owned Car Mart
Base Salary
$100
$150
$200
Dave
Sandi
Maureen
Dorothy
Function?: ______
Domain: ______________________
2) Employees of Sara’s Pre-Owned Car Mart
Phone Numbers
Dave
Sandi
Maureen
Dorothy
Function?: ______
Range: ____________________
555-2345
549-9402
930-3956
555-8294
839-9013
Domain: ______________________
3) (1, 4), (2,5), (3, 6), (4, 7)
5) (3,9), (2, 4), (0, 0), (1,1), ( 3,8)
Range: ____________________
4) (1, 4), (2, 4), (3,5), (6,10)
6) x   y  5
7) x 2  y  4
8) x2  y 2  4
9)
10)
Honors Precalculus
7
Name
Unit 1: Functions and Their Inverses
Date: _____________________
Warm-Up: 2
1. What is the difference quotient?
Find the difference quotient using the following functions.
2. f  x    6 x  2 x
2
4.
f  x 
2 x  1
x2  5
3. f ( x)  2 x  3x  6
2
5. f  x  
3x  1
x7
Honors Precalculus
8
Name
Unit 1: Functions and Their Inverses
Date: _____________________
Section 2.1 Domain – Part 1
Even Roots
Fractions
Logarithms
Even Roots in Fractions
Negative Exponents
Try These!
1) f  x  
2x
x 8
2) f  x  
3) f  x   log  x  4 
 x2  4 
5) f  x   

 x 1 
3
5 x  20
1
4) f  x    30  6 x  2
1
6) f  x   ln  6  2 x 
Honors Precalculus
9
Name
Unit 1: Functions and Their Inverses
Date: _____________________
State the domain of the following functions.
1.
f (x) = x 4 + 2x 2
3.
f (x) = x -1
4. f (x) =
5.
f ( x)  log  3x  2 
6.
7.
f (x) = 3x 3 - 2x + 1
8. f (x) = 7x + 21
9.
f (x) = (3x )
æ 2x +5 ö
10. f (x) = ç
÷
è 3x ø
1
-4
2.
f (x) = -2x -1
6x
x-4
f (x) = 3 4x +12
(
)
3
2
-1
Honors Precalculus
10
Name
Unit 1: Functions and Their Inverses
11.
f (x) = (5x +15)
13.
f (x) =
-
1
4
3x
x + 6x - 27
2
Date: _____________________
f (x) = -10x 2 + x - 3
14.
f (x) =
-5
15.
æ 3x +1 ö
f (x) = ç 2
÷
è x + 4x -12 ø
1
12.
16.
10
5
(
4x - 28
)
f (x) = x 2 +8x +15
-2
Honors Precalculus
11
Name
Unit 1: Functions and Their Inverses
Date: _____________________
Warm-Up: 3
Find the difference quotient using the following functions.
1)
f  x    5x2  7 x  4
2) f  x   9 
1 2
x
2
Use the given graph of the function to answer the following questions.
Honors Precalculus
Unit 1: Functions and Their Inverses
12
Name
Date: _____________________
Honors Precalculus
13
Name
Unit 1: Functions and Their Inverses
Date: _____________________
Section 2.1: Domain- Part II:
Graph the following functions and determine the domain.
1) f  x  
4
x 2  3x  10
2) f  x  
x2
6 x
 x3

 x5
3) f  x   log 
4) f  x  
x 1
x5
Notes: Domain – Part 2
Even Roots of Quadratics
Even Roots of Fractions
Logarithms with Fractions
State the domain of the following functions.
1) f  x  
4
x 2  3x  10
2) f  x  
x2
6 x
Honors Precalculus
14
Name
Unit 1: Functions and Their Inverses
 x3

 x5
3) f  x   log 
Date: _____________________
4) f  x  
x 1
x5
Try These!
State the domain of the following functions.
5) f  x  
4
x2
6) f  x  
 7 

 x 1 
8) f  x   log 
5 x
x4
2
10) f  x   3x  17 x  20
7) f  x   ln 
9) f  x  
2x2  x  3
 2 x 

 x4



1
6
Honors Precalculus
15
Name
Unit 1: Functions and Their Inverses
Date: _____________________
A) What are the “special cases” we need to watch out for when finding domain?
B) List the steps to follow when finding the domain of these “special cases”.
State the domain of the following functions.
1.
f (x) = 2x 2 -8
3.
f (x) = (12x - 48)
5.
f (x) =
3
5
x +1
8x -56
-1
12x
x -9
2.
f (x) =
4.
f (x) = 4 x 2 + 7x +12
6.
 x7 
f ( x)  log 

 2x  6 
2
Honors Precalculus
16
Name
Unit 1: Functions and Their Inverses
Date: _____________________
x2
5 x
7.
f (x) = 3 x 2+14x+ 24
8.
9.
f (x) =10 x 2 - 9x - 36
10.
9 x 
f ( x)  ln 

 2 
12.
 x  4 2
f ( x)  

 9 x  18 
14.
 10  2 x 
f ( x)  log 

 x3 
11.
13.
f (x) =
1
7x + 6
3
5x - 45
(
)
f (x) = x 2 + 8x
f ( x) 
1
4
Honors Precalculus
Unit 1: Functions and Their Inverses
Domain of Function: Final Summary
17
Name
Date: _____________________
Honors Precalculus
18
Name
Unit 1: Functions and Their Inverses
Date: _____________________
State the domain of the following functions.
2.
 x  9 
g ( x)  log 

 7 
3. h( x)  ln 
4.
j ( x) 
5.
6.
g ( x)   3  6 x  2
1.
f ( x) 
2x
x 8
 4x 1 

 10  x 
f ( x)  4 2 x 2  5 x  12


x2  9
7. h( x)   2

 6 x  13 x  5 
11  x
3 x  11x  4
2
1
1
8.
g ( x) 
6  3x
2x 1
Honors Precalculus
19
Name
Unit 1: Functions and Their Inverses
Date: _____________________
Section 2.3: Introduction to Intervals of Increasing, Decreasing and Constant…..
Use the graph to answer the following:
g(x)
f(x)
1) g 1  f  2 
Introduction to intervals of
increasing/decreasing/constant,
maximum and minimum.
f(x)
2) g  3  f  1
Maximum
Minimum
3) f  2   g  4 
4) f  3  g  0 
Increasing
Decreasing
Constant
5) Is f  2  positive or negative?
Domain
Range
6) Is g  5  positive or negative?
g(x)
Maximum
Minimum
Increasing
Decreasing
Constant
Domain
Range
Honors Precalculus
20
Name
Unit 1: Functions and Their Inverses
Date: _____________________
For each of the following graphs below:
(a) Find the domain and range
(b) Find the open domain intervals for which the function is
increasing/decreasing/constant. Label your answers clearly and If there is
none, write “none”.
(c) Find the minimum and maximum points on the function.
1.
(a) ______________________________
______________________________
(b) ______________________________
______________________________
______________________________
(c) ______________________________
______________________________
______________________________
2.
(a) ______________________________
______________________________
(b) ______________________________
______________________________
______________________________
(c) ______________________________
______________________________
______________________________
3.
(a) ______________________________
______________________________
(b) ______________________________
______________________________
______________________________
(c) ______________________________
______________________________
______________________________
Honors Precalculus
21
Name
Unit 1: Functions and Their Inverses
Date: _____________________
Warm-Up: 4
1. Draw a graph with the following characteristics:
y
10
9
8
1 minimum
7
6
5
2 intervals increasing
4
3
2
1 interval decreasing
1
x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6
7
8
9 10
1
2
3
4
5
6
7
8
9 10
-2
2 intervals constant
-3
-4
-5
-6
-7
-8
-9
-10
2. Draw a graph with the following characteristics:
y
10
9
8
2 intervals decreasing
7
6
5
3 intervals constant
4
3
2
Domain:  ,3   5,10
1
x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
3. Give the intervals of increasing, decreasing and constant.
a.
b.
Increasing
Increasing
___________________
___________________
Decreasing
Decreasing
___________________
____________________
Constant
Constant
___________________
____________________
Honors Precalculus
Unit 1: Functions and Their Inverses
22
Name
Date: _____________________
Honors Precalculus
23
Name
Unit 1: Functions and Their Inverses
Date: _____________________
Section 2.3: Even and Odd Functions
Determine Algebraically whether a Function is:
Even, Odd or Neither
No
Yes
Not Even
Even
(Stop!!)
Yes
Odd
(Stop!!)
A function is Even if:
Examples:
1
g  x  2
x
No
Neither
(Stop!!)
A function is Odd if:
 x3
h  x  2
x 9
Honors Precalculus
24
Name
Unit 1: Functions and Their Inverses
Date: _____________________
Determine algebraically if the function is even, odd, or neither.
1) f  x   3 x  5
2
3) f  x  
x
x 1
2
2) g  x   2 x  3x  6
2
4) g  x  
x
2x
Honors Precalculus
25
Name
Unit 1: Functions and Their Inverses
Date: _____________________
Determine whether each table represents an even or odd function or neither.
1) _____________________________________
x
-12
-9
-5
5
9
12
f(x)
7
-2
1
-1
2
-7
2) _____________________________________
x
-4
-3
-2
2
3
4
f(x)
10
8
6
6
8
10
3) _____________________________________
x
-8
-7
-6
6
7
8
f(x)
8
7
-6
6
7
8
4) _____________________________________
x
2
-5
6
5
-2
-6
f(x)
4
9
-7
-9
-4
7
5) _____________________________________
x
-10
8
-6
-8
10
6
f(x)
20
15
10
15
20
10
Honors Precalculus
26
Name
Unit 1: Functions and Their Inverses
Date: _____________________
Determine algebraically if the function is even, odd, or neither.
7. f ( x)  3x  x  5
8. f ( x)  x  2
4
2
9. f ( x )  x  2 x
10. f ( x)  x  5x  5
2
11. f ( x )  6 x
4
3
3
2x2
12. f ( x )  4
5x  1
State the domain.
13. f (x) 
2 x  x  15
2
14. f ( x) 
x2  4
x8
Honors Precalculus
27
Name
Unit 1: Functions and Their Inverses
Date: _____________________
Warm-Up: 5
Determine whether the graph of the function is even or odd function or neither. Briefly
explain your answer.
1.
2.
____________________________
3.
________________________
________________________
Determine whether each table represents an even or odd function or neither.
4. _____________________________________
x
-7
-5
-3
3
5
7
f(x)
-4
-2
-1
1
2
4
5. _____________________________________
x
-6
-4
-2
2
4
6
f(x)
8
-5
-1
1
5
-8
6. _____________________________________
x
-5
-3
-1
1
3
5
f(x)
10
5
4
4
-5
10
Honors Precalculus
28
Name
Unit 1: Functions and Their Inverses
Date: _____________________
Section 2.2 and Section 2..4: Graphs of Common Functions and Their Transformations
Sketch an accurate graph for each of the following parent functions, do your best without
the calculator. Determine whether the graph is an even or odd function, or neither.
The Constant Function
f ( x)  c for any real number c
The Cubic Function
The Linear Function
f ( x) 
1
x
f ( x)  x 2
f ( x)  x
The Square Root Function
f ( x)  x 3
The Reciprocal Function
The Quadratic Function
f ( x) 
f ( x) 
x
The Absolute Value Function
f ( x)  x
The Cube Root Function
3
x
The Exponential Function
f ( x)  a x for any real #a, a  1
Honors Precalculus
29
Name
Unit 1: Functions and Their Inverses
Date: _____________________
1) Why can’t you use all x- values as key points for a square root
function?
2) By looking at a graph, how do you tell whether it is an odd or an even
function?
Graph the following transformations. List the transformations from the parent
graph, fill in the chart and sketch the graph.
3) f (x) = 3 x + 2
y
10
9
8
7
6
5
4
3
2
1
x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
(
)
2
4) g(x) = x -1 - 3
1
2
3
4
5
6
7
8
9 10
Honors Precalculus
Unit 1: Functions and Their Inverses
()
5) h x =
()
6) j x =
x +1 - 2
3
2
+1
x-3
()
7) f x = 2
()
8) g x = - x + 3
2
30
Name
-x
Date: _____________________
Honors Precalculus
31
Name
Unit 1: Functions and Their Inverses
Date: _____________________
Notes- Find the equation of a function given a graph
a) Identify the parent function and describe the transformation shown in the graph. b)
Write an equation for the graphed function. c) State the domain and range of the
function.
1.
3.
2.
Honors Precalculus
32
Name
Unit 1: Functions and Their Inverses
4.
6.
Date: _____________________
5.
Honors Precalculus
33
Name
Unit 1: Functions and Their Inverses
Date: _____________________
Section 2.4: Piecewise Functions
ì 3x - 5, x > 4
Evaluate the following for f ( x ) = í
2
îx ,
1) f ( 7 )
x£4
.
2) f ( 4 )
3) f ( -3)
ì - 2 x +1 , x £1
ï
Evaluate the following for f ( x ) = í 3,
1 < x < 3.
ï6 - 2x,
x³3
î
4) f (10 )
5) f ( 2 )
6) f ( 0 )
Graph the following piecewise functions.
ì - 2, x < 0
x³0
î 3,
7) f ( x ) = í
8)
ì - x + 2,
f ( x) = í
î x - 2,
x<2
x³2
y
y
10
10
9
9
8
8
7
7
6
6
5
5
4
4
3
3
2
2
1
1
x
x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6
7
8
9 10
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
-2
-2
-3
-3
-4
-4
-5
-5
-6
-6
-7
-7
-8
-8
-9
-9
-10
-10
1
2
3
4
5
6
7
8
9 10
Honors Precalculus
34
Name
Unit 1: Functions and Their Inverses
ì - 3x + 2,
ï
9) f ( x ) = í 1
ïî 2 x - 4,
Date: _____________________
ì 4,
ï
10) f ( x ) = í x 2 ,
ï 4,
î
x£2
x>2
x £ -2
-2< x <2
x³2
y
y
10
10
9
9
8
8
7
7
6
6
5
5
4
4
3
3
2
2
1
1
x
x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6
7
8
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
9 10
-2
-2
-3
-3
-4
-4
-5
-5
-6
-6
-7
-7
-8
-8
-9
-9
-10
-10
x £ -3
ì 3x +12,
ï
11) f ( x ) = í x ,
-3< x <3
ï -3x +12,
x³3
î
1
2
3
4
5
6
7
8
9 10
8
9 10
ì x 2 - 4, x < 3
ï
12) f ( x ) = í 2
ï 3 x - 5, x ³ 3
î
y
y
10
10
9
9
8
8
7
7
6
6
5
5
4
4
3
3
2
2
1
1
x
x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6
7
8
9 10
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
-2
-2
-3
-3
-4
-4
-5
-5
-6
-6
-7
-7
-8
-8
-9
-9
-10
-10
1
2
3
4
5
6
7
Honors Precalculus
35
Name
Unit 1: Functions and Their Inverses
Date: _____________________
Warm-Up: 6
Given f  x  
4x
and
g  x 
x2
. Perform the indicated operations. State
x 1
the domain for parts a-c.
a)
f
 g  x 
b)
g
c)
f
 g  x 
d) 
e)
 g  f  2 
 f  x 
f 
  x
g
 f 
  0
g
f) 
Honors Precalculus
Unit 1: Functions and Their Inverses
36
Name
Date: _____________________
Honors Precalculus
37
Name
Unit 1: Functions and Their Inverses
Date: _____________________
Composition of Functions:
Given the following functions answer parts a-c.
1. Given:
a.
f ( x)  4 x 2  x
a. f ( g (1))
2
x3
c.
g
f
 x 
c.
f
g  x 
g( x)  2 x  5 , find
b. g ( f ( x))
g ( f ( 2))
3. Given: f ( x) 
g( x)  x 2  3 , find
b. f ( g ( x))
f ( g (2))
2. Given:
a.
f ( x)  3x  8
g ( x)  5 x  8 , find
b.
f
f  x 
c.
g
g  x 
Honors Precalculus
38
Name
Unit 1: Functions and Their Inverses
Date: _____________________
Domain of a Composition of Functions
Examples:
4. Given f  x  
a. Find
g
f
 x  .
5. Given f  x  
a. Find
g
f
g
g  x  .
f
f
1
3 x
 x  .
6. Given f  x  
a. Find
3
and g  x  
x 1
x
2
b.
f
g  x 
and g  x  
b.
f
c. Find the domain of
f
g  x  .
c. Find the domain of
f
g  x  .
x2  2
g  x 
x and g  x   x 2  1
 x  .
b.
f
g  x 
c. Find the domain of
Honors Precalculus
39
Name
Unit 1: Functions and Their Inverses
Date: _____________________
Graph the following function. List all transformations. State the domain and range.
1. f ( x )  23 x  1  4
2.
f ( x)  
1
x  3 1
2
y
y
10
10
9
9
8
8
7
7
6
6
5
5
4
4
3
3
2
2
1
1
x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6
7
8
9 10
x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
-2
-2
-3
-3
-4
-4
-5
-5
-6
-6
-7
-7
-8
-8
-9
-9
-10
-10
Find the function of the given graphs.
3.
4.
1
2
3
4
5
6
7
8
9 10
Honors Precalculus
40
Name
Unit 1: Functions and Their Inverses
Date: _____________________
Describe the transformations from each parent function f  x  to the function g  x  .
5.
f  x 
6.
f ( x)  x ;
g ( x)   4 x  6  8
7.
f ( x)  x 2 ;
g ( x) 
8.
f ( x) 
x;
3
g ( x) 
x;
3
x4  3
2
1
2
 x  3  2
2
g ( x)  
3
x5 6
Write a function based on the transformations that are given.
9. An absolute value function that is shifted 3 units to the right, reflected over the xaxis, and vertically shrunk by
1
.
4
10. A reciprocal function is shifted 2 units left, down 4 units, and vertically stretched
by a factor of 3.
11. A cubic function is reflected over the y- axis, stretched by a factor of 2 and moved
up 4.
Honors Precalculus
41
Name
Unit 1: Functions and Their Inverses
1) If f ( x ) 
Date: _____________________
2x  3
4x
and g ( x ) 
, find the following functions and state the domain of
3x  2
3x  2
each.
a) ( f  g )( x)
b) ( f  g )( x)
c) ( f  g )( x)
d) 
 f 
 ( x)
g
#2-4
(a) Find f g .
2)
f ( x) 
g ( x) 
2
x
x
(b) Find the domain of f g .
and
g ( x)  2 x  3
3) f ( x) 
3
x 1
and
Honors Precalculus
Unit 1: Functions and Their Inverses
4) f ( x) 
x 1
42
Name
and
g ( x) 
Date: _____________________
2
x 1
Use the graphs of f , g and h to evaluate the functions.
f ( x)
g ( x)
h ( x)
5. ( f  g )(2)
6. ( f  g )(1)
7. ( f  g )(0)
8. ( f h)(3)
9. ( g
10. ( f
11. ( f
12. ( g h)(0)
13. ( g g g )(1)
14.
f )(4)
(h f g )(2)
g h)(2)
f )(1)
Honors Precalculus
43
Name
Unit 1: Functions and Their Inverses
Date: _____________________
Section 5.1: Working Backwards, Decomposing Functions
Find two functions, f  x  and g  x  , so that H  x    f
1) H  x  
1
x 1
2) H  x  
2
 x  3
5) h  x  
4) h  x   5 x  6
g  x  .
3) H  x  
2
1
x 1
3
x2  4
6) h  x   x 
2
2
7) Find three functions whose composition is h  x  
x2  1  3
1
x
. Try to find four?
Five?
8) For each of the functions f  x  and h  x  below, find a function g  x  such that
h( x ) = ( f g )( x ) .
x
(a) f  x   10 x , h  x   10
2
17

(b) f  x  

(c) f  x   x , h  x   sin  x 
3

3
x , h  x 
x2  4
Honors Precalculus
Unit 1: Functions and Their Inverses
44
Name
Date: _____________________
Honors Precalculus
45
Name
Unit 1: Functions and Their Inverses
Date: _____________________
Warm-Up: 7
1) Given f  x  
3
x 4
2
and
g  x 
f 
  x
g
a) 
x  5 . Find the following and give the domain.
b)
f
2) Find two functions, f  x  and g  x  , so that H  x    f
a) H  x   2 x  1  4
3) Given f  x  
a)
f
2x  5
x4
 g  x 
b) H  x  
and
g  x 
g  x 
g  x  .
x5
x 1
3 x
. Find the following and give the domain.
x4
g
b)    x 
c) f  g  x  
f 
Honors Precalculus
46
Name
Unit 1: Functions and Their Inverses
Date: _____________________
4) Use the tables below to find the following:
x
-4
-2
-1
0
2
5
f(x)
3
6
9
-6
-2
0
x
-9
-6
-2
0
3
10
g(x)
7
0
4
-2
1
5
a) f  g  0  
b) g  f  4  
c) g  f  2  
d) g  f  5  
e) f  g  6  
f) f  g 10  
Honors Precalculus
47
Name
Unit 1: Functions and Their Inverses
Date: _____________________
Section 5.2: Inverses
Introduction
Properties of Inverses

The inverse function for f(x), labeled f−1(x) (which is read “ f inverse of x”),
contains the same domain and range elements as the original function, f( x).
However, the sets are switched.

In other words, the domain of f( x) is the range of f−1( x), and vice versa.

In fact, for every ordered pair ( a, b) belonging to f( x), there is a corresponding
ordered pair ( b, a) that belongs to f−1( x).
For example, consider this function, g:
g :{(2, 0), (1,3), (5,9)}
The inverse function is the set of all ordered pairs reversed:
g 1 :{(0, 2), (3,1), (9,5)}
Graphs of inverse functions
Since functions and inverse functions contain the same numbers in their ordered pair,
just in reverse order, their graphs will be reflections of one another across the line y = x.
Figure 1 Inverse functions are symmetric about the line y = x.
Honors Precalculus
Unit 1: Functions and Their Inverses
48
Name
Date: _____________________
Part I) Given:
f ( x)
Ordered Pairs of f ( x)
f 1 ( x )
1
Ordered Pairs of f ( x )
Inverse Notation:
An inverse………..
Verify Algebraically
f  f 1 ( x) 
f 1  f ( x) 
Honors Precalculus
49
Name
Unit 1: Functions and Their Inverses
Date: _____________________
Part II) For a function to have an inverse that is a function, it must be one-to-one.
How can we make this function 1 to 1?
f (x) = x 2
All functions have inverses, but not all inverses of functions are functions.
Find the inverse:
1) f (x) =
3x - 5
2x +1
2) f (x) =
4x +1
x +10
Honors Precalculus
50
Name
Unit 1: Functions and Their Inverses
Date: _____________________
Part III) Try These!
#1-3:
Determine whether the given function is 1-1. If it is, give the domain and
range of the inverse of the following functions. You may abbreviate names using letters.
1)
DOMAIN
RANGE
Employees of Sara’s
Base Salary
Pre-Owned Car Mart
Jim
$100
Paula
$150
Bill
$200
Mary
Laura
2) {(-3,-27), (-2,-8), (-1,-1), (0,0), (1,1,), (2,8), (3,27)}
3) { (-3,9), (-2,4), (-1,1), (0,0), (1,1), (2,4), (3,9)}
Complete the following:
4) If f (1)  5, f (3)  7, and f (8)   10 , find f
1
(7), f 1 (5), and f -1 (-10) .
5) If the point  3, 5  is on the graph of a one-to-one function f , what point must be
on the graph of f
1
?
Honors Precalculus
51
Name
Unit 1: Functions and Their Inverses
1) Use the function f ( x) 
Date: _____________________
x  4  2 to answer the questions that follow. Show all work
in the space provided and label answers clearly.
a. Is f  x  1 to 1? _______________________________
If it is not 1 to 1, restrict the domain so that it is.
____________________________ (Use your answer to a for parts b to i)
b. List the parent graph, and all transformations from the parent graph.
Parent function: _________________________________
Transformations (in words):______________________________________________________________
c. Graph f  x  on the graph provided. List 3 key points on the lines provided.
y
10
Key Points on f  x  :
9
__________
___________
__________
8
7
6
5
d. State the domain of f  x  .
4
_______________________
3
2
1
x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
e. State the range of f  x  . _______________________
1
-2
-3
-4
f. Find f 1  x  algebraically. Show all work.
____________________
-5
-6
-7
-8
-9
-10
g. Graph f 1  x  on the same graph. List the 3 key points on the lines provided .
Key points on f 1  x  :
___________
h. State the domain of f 1  x  .
___________
_______________________
i. State the range of f 1  x  . _______________________
__________
2
3
4
5
6
7
8
9 10
Honors Precalculus
52
Name
Unit 1: Functions and Their Inverses
Date: _____________________
Use the function f ( x)  2  x  3  1 to answer the questions that follow. Show all work in
2
the space provided and label answers clearly.
a. Is f  x  1 to 1? _______________________________
If it is not 1 to 1, restrict the domain so that it is.
____________________________ (Use your answer to a for parts b to i)
b. List the parent graph, and all transformations from the parent graph.
Parent function: _________________________________
Transformations (in words):______________________________________________________________
c. Graph f  x  on the graph provided. List 3 key points on the lines provided.
Key Points on f  x  :
__________
d. State the domain of f  x  .
___________
__________
_______________________
e. State the range of f  x  . _______________________
f. Find f 1  x  algebraically. Show all work.
____________________
g. Graph f 1  x  on the same graph. List the 3 key points on the lines provided .
Key points on f 1  x  :
___________
h. State the domain of f 1  x  .
___________
_______________________
i. State the range of f 1  x  . _______________________
__________
Honors Precalculus
53
Name
Unit 1: Functions and Their Inverses
Date: _____________________
Warm-Up: 8
1. What is the definition of a one-to-one function?
Are the following functions one-to-one?
2.
3.
X -1 -0.5 0 0.5 3
X -2 -1 0 1 2
Y 1
Y 3
2
1
6
0
4.. f ( x)  2 x  3x  8
5
7 9 11
5. 4 x  2 y  6
2
Find the inverse of the following functions.
bg b g
6. f x   x  1  8
Verify that f ( x ) and f
8. f ( x)   5 x  3
7. f ( x )  43 x  2  5
2
1
b xg are inverses.
3 1
g ( x)   x
5 5
9. f ( x)  4 3x  6
g ( x) 
1 4
x 2 , x0
3