Functions

Chapter
5
Polynomials
5.1
Functions
1. Consider again the relation R.
R = {(7, 4), (2, 4), (4, 2), (8, 5)}
To form the domain, we take the first element of each ordered pair and put it
into a set.
{7, 2, 4, 8}
However, in listing the final answer, we should eliminate duplicate elements
and then sort the numbers in numerical order, from smallest to largest.
Domain = {2, 4, 7, 8}
To find the range, we take the second element of each ordered pair and put it
in a set.
{4, 4, 2, 5}
However, in listing the final answer, we should eliminate duplicate elements
and then sort the numbers in numerical order, from smallest to largest.
Range = {2, 4, 5}
3. Consider again the relation T.
T = {(7, 2), (3, 1), (9, 4), (8, 1)}
To form the domain, we take the first element of each ordered pair and put it
into a set.
{7, 3, 9, 8}
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270
However, in listing the final answer, we should eliminate duplicate elements
and then sort the numbers in numerical order, from smallest to largest.
Domain = {3, 7, 8, 9}
To find the range, we take the second element of each ordered pair and put it
in a set.
{2, 1, 4, 1}
However, in listing the final answer, we should eliminate duplicate elements
and then sort the numbers in numerical order, from smallest to largest.
Range = {1, 2, 4}
5. Consider again the relation T.
T = {(4, 7), (4, 8), (5, 0), (0, 7)}
To form the domain, we take the first element of each ordered pair and put it
into a set.
{4, 4, 5, 0}
However, in listing the final answer, we should eliminate duplicate elements
and then sort the numbers in numerical order, from smallest to largest.
Domain = {0, 4, 5}
To find the range, we take the second element of each ordered pair and put it
in a set.
{7, 8, 0, 7}
However, in listing the final answer, we should eliminate duplicate elements
and then sort the numbers in numerical order, from smallest to largest.
Range = {0, 7, 8}
7. Consider again the relation given graphically.
y
5
A
B
x
−5
5
C
D
−5
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Note the coordinates of each point: A = (2, 4), B = (−2, 2), C = (−2, −2), and
D = (2, −2). To form the domain, we take the first element of each ordered
pair and put it into a set.
{2, −2, −2, 2}
However, in listing the final answer, we should eliminate duplicate elements
and then sort the numbers in numerical order, from smallest to largest.
Domain = {−2, 2}
To find the range, we take the second element of each ordered pair and put it
in a set.
{4, 2, −2, −2}
However, in listing the final answer, we should eliminate duplicate elements
and then sort the numbers in numerical order, from smallest to largest.
Range = {−2, 2, 4}
9. Consider again the relation given graphically.
y
B
5
A
x
−5
5
C
D
−5
Note the coordinates of each point: A = (1, 2), B = (−1, 4), C = (−4, −2), and
D = (2, −2). To form the domain, we take the first element of each ordered
pair and put it into a set.
{1, −1, −4, 2}
However, in listing the final answer, we should eliminate duplicate elements
and then sort the numbers in numerical order, from smallest to largest.
Domain = {−4, −1, 1, 2}
To find the range, we take the second element of each ordered pair and put it
in a set.
{2, 4, −2, −2}
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However, in listing the final answer, we should eliminate duplicate elements
and then sort the numbers in numerical order, from smallest to largest.
Range = {−2, 2, 4}
11. Consider again the relation.
R = {(−6, −4), (−4, −4), (1, −4)}
List the elements of the domain on the left, the elements of the range on the
right, then use arrows to indicate the connection between the first and second
elements of each ordered pair.
R
−6
−4
1
−4
Note that each domain element is paired with exactly one range element.
Hence, R is a function.
13. Consider again the relation.
T = {(−1, −7), (2, −5), (4, −2)}
List the elements of the domain on the left, the elements of the range on the
right, then use arrows to indicate the connection between the first and second
elements of each ordered pair.
T
−1
2
4
−7
−5
−2
Note that each domain element is paired with exactly one range element.
Hence, T is a function.
15. Consider again the relation.
T = {(−9, 1), (1, 6), (1, 8)}
List the elements of the domain on the left, the elements of the range on the
right, then use arrows to indicate the connection between the first and second
elements of each ordered pair.
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T
−9
1
1
6
8
Note that the domain element 1 is paired with two range elements, 1 and 6.
Hence, the relation T is not a function.
17. Consider again the relation.
R = {(−7, −8), (−7, −6), (−5, 0)}
List the elements of the domain on the left, the elements of the range on the
right, then use arrows to indicate the connection between the first and second
elements of each ordered pair.
R
−7
−5
−8
−6
0
Note that the domain element −7 is paired with two range elements, −8 and
−6. Hence, the relation R is not a function.
19. Consider again the relation.
y
5
C
B
A
−1
−1
5
x
Create a mapping diagram for the points A = (1, 2), B = (2, 3), and C = (4, 4).
List the elements of the domain on the left, the elements of the range on the
right, then use arrows to indicate the connection between the first and second
elements of each ordered pair.
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1
2
4
2
3
4
Note that each domain element is paired with exactly one range element.
Hence, the relation is a function.
21. Consider again the relation.
y
5
C
B
A
−1
−1
5
x
Create a mapping diagram for the points A = (1, 1), B = (1, 2), and C = (2, 4).
List the elements of the domain on the left, the elements of the range on the
right, then use arrows to indicate the connection between the first and second
elements of each ordered pair.
1
2
1
2
4
Note that the domain element 1 is paired with two range elements, 1 and 2.
Hence, the relation is not a function.
23. Given f (x) = |6x − 9|, to evaluate f (8), first restate the function notation,
then replace each occurrence of the variable with open parentheses.
f (x) = |6x − 9|
Original function notation.
f ( ) = |6( ) − 9|
Replace each occurrence of x with
open parentheses.
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Now substitute 8 for x in the open parentheses prepared in the last step.
f (8) = |6(8) − 9|
Substitute 8 for x in the open
parentheses positions.
f (8) = |48 − 9|
f (8) = |39|
Multiply: 6(8) = 48.
Simplify.
f (8) = 39
Take absolute value.
Hence, f (8) = 39; i.e., f sends 8 to 39.
25. Given f (x) = −2x2 +8, to evaluate f (3), first restate the function notation,
then replace each occurrence of the variable with open parentheses.
f (x) = −2x2 + 8
2
f ( ) = −2( ) + 8
Original function notation.
Replace each occurrence of x with
open parentheses.
Now substitute 3 for x in the open parentheses prepared in the last step.
f (3) = −2(3)2 + 8
Substitute 3 for x in the open
parentheses positions.
f (3) = −2(9) + 8
f (3) = −18 + 8
Exponent first: (3)2 = 9
Multiply: −2(9) = −18
and 0(3) = 0
f (3) = −10
Simplify.
Hence, f (3) = −10; i.e., f sends 3 to −10.
27. Given f (x) = −3x2 + 4x + 1, to evaluate f (2), first restate the function
notation, then replace each occurrence of the variable with open parentheses.
f (x) = −3x2 + 4x + 1
2
f ( ) = −3( ) + 4( ) + 1
Original function notation.
Replace each occurrence of x with
open parentheses.
Now substitute 2 for x in the open parentheses prepared in the last step.
f (2) = −3(2)2 + 4(2) + 1
Substitute 2 for x in the open
parentheses positions.
f (2) = −3(4) + 4(2) + 1
f (2) = −12 + 8 + 1
Exponent first: (2)2 = 4
Multiply: −3(4) = −12
and 4(2) = 8
Simplify.
f (2) = −3
Hence, f (2) = −3; i.e., f sends 2 to −3.
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29. Given f (x) = |5x+9|, to evaluate f (−8), first restate the function notation,
then replace each occurrence of the variable with open parentheses.
f (x) = |5x + 9|
f ( ) = |5( ) + 9|
Original function notation.
Replace each occurrence of x with
open parentheses.
Now substitute −8 for x in the open parentheses prepared in the last step.
f (−8) = |5(−8) + 9|
f (−8) = | − 40 + 9|
Substitute −8 for x in the open
parentheses positions.
Multiply: 5(−8) = −40.
f (−8) = | − 31|
f (−8) = 31
Simplify.
Take absolute value.
Hence, f (−8) = 31; i.e., f sends −8 to 31.
√
31. Given f (x) = x − 6, to evaluate f (42), first restate the function notation,
then replace each occurrence of the variable with open parentheses.
√
Original function notation.
f (x) = x − 6
f( ) = ( ) − 6
Replace each occurrence of x with
open parentheses.
Now substitute 42 for x in the open parentheses prepared in the last step.
Substitute 42 for x in the open
f (42) = (42) − 6
parentheses positions.
√
f (42) = 36
Simplify.
√
f (42) = 6
Take square root: 36 = 6
Hence, f (42) = 6; i.e., f sends 42 to 6.
√
33. Given f (x) = x − 7, to evaluate f (88), first restate the function notation,
then replace each occurrence of the variable with open parentheses.
√
Original function notation.
f (x) = x − 7
Replace each occurrence of x with
f( ) = ( ) − 7
open parentheses.
Now substitute 88 for x in the open parentheses prepared in the last step.
Substitute 88 for x in the open
f (88) = (88) − 7
parentheses positions.
√
f (88) = 81
Simplify.
√
f (88) = 9
Take square root: 81 = 9
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Hence, f (88) = 9; i.e., f sends 88 to 9.
35. Given f (x) = −4x + 6, to evaluate f (8), first restate the function notation,
then replace each occurrence of the variable with open parentheses.
f (x) = −4x + 6
Original function notation.
f ( ) = −4( ) + 6
Replace each occurrence of x with
open parentheses.
Now substitute 8 for x in the open parentheses prepared in the last step.
f (8) = −4(8) + 6
Substitute 8 for x in the open
parentheses positions.
f (8) = −32 + 6
f (8) = −26
Multiply: −4(8) = −32
Add: −32 + 6 = −26
Hence, f (8) = −26; i.e., f sends 8 to −26.
37. Given f (x) = −6x + 7, to evaluate f (8), first restate the function notation,
then replace each occurrence of the variable with open parentheses.
f (x) = −6x + 7
f ( ) = −6( ) + 7
Original function notation.
Replace each occurrence of x with
open parentheses.
Now substitute 8 for x in the open parentheses prepared in the last step.
f (8) = −6(8) + 7
f (8) = −48 + 7
Substitute 8 for x in the open
parentheses positions.
Multiply: −6(8) = −48
f (8) = −41
Add: −48 + 7 = −41
Hence, f (8) = −41; i.e., f sends 8 to −41.
39. Given f (x) = −2x2 + 3x + 2 and g(x) = 3x2 + 5x − 5, to evaluate f (3),
first choose the function f , then replace each occurrence of the variable with
open parentheses.
f (x) = −2x2 + 3x + 2
2
f ( ) = −2( ) + 3( ) + 2
Original function notation.
Replace each occurrence of x with
open parentheses.
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Now substitute 3 for x in the open parentheses prepared in the last step.
f (3) = −2(3)2 + 3(3) + 2
Substitute 3 for x in the open
parentheses positions.
f (3) = −2(9) + 3(3) + 2
Exponent first: (3)2 = 9
f (3) = −18 + 9 + 2
Multiply: −2(9) = −18
and 3(3) = 9
f (3) = −7
Simplify.
Hence, f (3) = −7; i.e., f sends 3 to −7. To evaluate g(3), repeat the same
procedure, this time using the function g.
g(x) = 3x2 + 5x − 5
2
g( ) = 3( ) + 5( ) − 5
Original function notation.
Replace each occurrence of x with
open parentheses.
Now substitute 3 for x in the open parentheses prepared in the last step.
g(3) = 3(3)2 + 5(3) − 5
Substitute 3 for x in the open
parentheses positions.
g(3) = 3(9) + 5(3) − 5
Exponent first: (3)2 = 9
g(3) = 27 + 15 − 5
Multiply: 3(9) = 27
and 5(3) = 15
g(3) = 37
Simplify.
Hence, g(3) = 37; i.e., g sends 3 to 37.
41. Given f (x) = 6x − 2 and g(x) = −8x + 9, to evaluate f (−7), first choose
the function f (x) = 6x − 2, then replace each occurrence of the variable with
open parentheses.
f (x) = 6x − 2
f ( ) = 6( ) − 2
Original function notation.
Replace each occurrence of x with
open parentheses.
Now substitute −7 for x in the open parentheses prepared in the last step.
f (−7) = 6(−7) − 2
f (−7) = −42 − 2
Substitute −7 for x in the open
parentheses positions.
Multiply: 6(−7) = −42
f (−7) = −44
Simplify.
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Hence, f (−7) = −44; i.e., f sends −7 to −44. Now, repeat the procedure,
using the function g.
g(x) = −8x + 9
g( ) = −8( ) + 9
Original function notation.
Replace each occurrence of x with
open parentheses.
Now substitute −7 for x in the open parentheses prepared in the last step.
g(−7) = −8(−7) + 9
g(−7) = 56 + 9
Substitute −7 for x in the open
parentheses positions.
Multiply: −8(−7) = 56
g(−7) = 65
Simplify.
Hence, g(−7) = 65; i.e., g sends −7 to 65.
43. Given f (x) = 4x−3 and g(x) = −3x+8, to evaluate f (−3), first choose the
function f , then replace each occurrence of the variable with open parentheses.
f (x) = 4x − 3
f ( ) = 4( ) − 3
Original function notation.
Replace each occurrence of x with
open parentheses.
Now substitute −3 for x in the open parentheses prepared in the last step.
f (−3) = 4(−3) − 3
f (−3) = −12 − 3
Substitute −3 for x in the open
parentheses positions.
Multiply: 4(−3) = −12
f (−3) = −15
Simplify.
Hence, f (−3) = −15; i.e., f sends −3 to −15. To evaluate g(−3), repeat the
same procedure, this time using the function g.
g(x) = −3x + 8
g( ) = −3( ) + 8
Original function notation.
Replace each occurrence of x with
open parentheses.
Now substitute −3 for x in the open parentheses prepared in the last step.
g(−3) = −3(−3) + 8
g(−3) = 9 + 8
Substitute −3 for x in the open
parentheses positions.
Multiply: −3(−3) = 9
g(−3) = 17
Simplify.
Hence, g(−3) = 17; i.e., g sends −3 to 17.
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