Solved Examples and Worksheet for Operations with Functions
Q1
Find a rule for the function
(
f
-
g
)(
x
),
if
f
(
x
) =
x
x
2
-
9
,
g
(
x
) =
3
x
2
-
9
.
A
. (
f
-
g
)(
x
) =
1
3
-
x
B
. (
f
-
g
)(
x
) =
-
1
x
+
3
C
. (
f
-
g
)(
x
) =
1
x
-
3
D
. (
f
-
g
)(
x
) =
1
x
+
3
Solutions
Share
Step: 1
f
(
x
) =
x
x
2
-
9
,
g
(
x
) =
3
x
2
-
9
[Original functions.]
Step: 2
(
f
-
g
)(
x
) =
f
(
x
) -
g
(
x
)
[Write the rule for the difference of functions.]
Step: 3
=
x
x
2
-
9
-
3
x
2
-
9
[Substitute the values for
f
(
x
) and
g
(
x
).]
Step: 4
=
x
-
3
x
2
-
9
[Subtract.]
Step: 5
=
x
-
3
(
x
-
3
)
(
x
+
3
)
[Factor.]
Step: 6
=
1
x
+
3
[Simplify.]
Step: 7
(
f
-
g
)(
x
) =
1
x
+
3
Correct Answer is :
(
f
-
g
)(
x
) =
1
x
+
3
Q2
If
f
(
x
) = 3
x
+ 6
and
g
(
x
) = 3
x
+ 9,
find
(
f
+
g
)(
x
).
A
. 6
x
+ 16
B
. 6
x
+ 15
C
. 6
x
+ 13
D
. 9
x
+ 12
E
. 6
x
+ 14
Solutions
Share
Step: 1
(
f
+
g
)(
x
) =
f
(
x
) +
g
(
x
).
Step: 2
= 3
x
+ 6 + 3
x
+ 9.
Step: 3
= 6
x
+ 15.
Correct Answer is :
6
x
+ 15
Q3
If
f
(
x
) =
x
2
- 15
x
+ 8,
g
(
x
) =
x
2
- 21
x
+ 10,
what is the value of
(
f
-
g
)(
x
)?
A
. 6
x
B
. 6
x
+ 2
C
. - 6
x
- 2
D
. - 6
x
+ 2
E
. 6
x
- 2
Solutions
Share
Step: 1
Given,
f
(
x
) =
x
2
- 15
x
+ 8 and
g
(
x
) =
x
2
- 21
x
+ 10.
Step: 2
(
f
-
g
)(
x
) =
f
(
x
) -
g
(
x
) =
x
2
- 15
x
+ 8 - (
x
2
- 21
x
+ 10)
Step: 3
=
x
2
- 15
x
+ 8 -
x
2
+ 21
x
- 10 = 6
x
- 2
Correct Answer is :
6
x
- 2
Q4
If
f
(
x
) = 2
x
,
g
(
x
) =
x
4
+ 4,
what is the value of
(
f
×
g
)(
x
)?
A
. 2
x
5
B
. 2
x
5
+ 8
x
C
. 2
x
+
x
4
+ 2
D
.
x
4
+ 8
E
. 2
x
4
+ 8
x
Solutions
Share
Step: 1
Given,
f
(
x
) = 2
x
and
g
(
x
) =
x
4
+ 4.
Step: 2
(
f
×
g
)(
x
) =
f
(
x
) ×
g
(
x
)
Step: 3
= 2
x
× (
x
4
+ 4) = 2
x
5
+ 8
x
Correct Answer is :
2
x
5
+ 8
x
Q5
If
f
(
x
) = 8
x
,
g
(
x
) =
x
2
,
what is the value of (
f
g
)(
x
)?
A
.
x
2
B
. 8
x
C
. 8
x
3
D
.
8
x
E
. 8
x
+
x
2
Solutions
Share
Step: 1
Given,
f
(
x
) = 8
x
and
g
(
x
) =
x
2
Step: 2
(
f
g
)(
x
) =
f
(
x
)
g
(
x
)
,
g
(
x
) ≠ 0
Step: 3
=
8
x
x
2
,
g
(
x
) ≠ 0
Step: 4
=
8
x
Correct Answer is :
8
x
Q6
If
f
(
x
) = - 2
x
2
+ 4
x
- 9 and
g
(
x
) =
x
2
+ 8, then find (
f
-
g
)(
x
).
A
. - 3
x
2
+ 4
x
- 17
B
. 3
x
2
+ 4
x
- 17
C
. -
x
2
+ 4
x
- 17
D
. - 3
x
2
+ 4
x
+ 17
Solutions
Share
Step: 1
(
f
-
g
)(
x
) =
f
(
x
) -
g
(
x
)
Step: 2
= - 2
x
2
+ 4
x
- 9 - (
x
2
+ 8)
[Substitute the values.]
Step: 3
= - 2
x
2
+ 4
x
- 9 -
x
2
- 8
Step: 4
= - 3
x
2
+ 4
x
- 17
[Simplify.]
Correct Answer is :
- 3
x
2
+ 4
x
- 17
Q7
If
f
(
x
) = - 7
x
5
-
x
2
+
x
+ 10 and
g
(
x
) = -
x
2
+
x
, then find (
f
-
g
)(
x
).
A
. 7
x
5
+ 10
B
. - 7
x
5
+ 2
x
2
+ 10
C
. - 7
x
5
+ 10
D
. - 7
x
5
+
x
+ 10
Solutions
Share
Step: 1
(
f
-
g
)(
x
) =
f
(
x
) -
g
(
x
)
Step: 2
= - 7
x
5
-
x
2
+
x
+ 10 - (-
x
2
+
x
)
[Substitute the values.]
Step: 3
= - 7
x
5
-
x
2
+
x
+ 10 +
x
2
-
x
Step: 4
= - 7
x
5
+ 10
[Simplify.]
Correct Answer is :
- 7
x
5
+ 10
Q8
Find (
f
-
g
)(
x
) for
f
(
x
) =
x
+ 3 and
g
(
x
) =
x
2
+
x
- 1.
A
. -
x
2
+ 4
B
.
x
2
- 4
C
. -
x
2
-
x
+ 3
D
. -
x
2
- 2
x
+ 4
Solutions
Share
Step: 1
(
f
-
g
)(
x
) =
f
(
x
) -
g
(
x
)
[Subtraction of functions.]
Step: 2
= (
x
+ 3) - (
x
2
+
x
- 1)
[
f
(
x
) =
x
+ 3 and
g
(
x
) =
x
2
+
x
- 1.]
Step: 3
= -
x
2
+ 4
[Simplify.]
Correct Answer is :
-
x
2
+ 4
Q9
If
f
(
x
) = 11
x
+ 3
and
g
(
x
) = 11
x
+ 7,
find
(
f
+
g
)(
x
).
A
. 6
x
+ 7
B
.
x
+ 10
C
. 121
x
+ 10
D
. 22
x
+ 10
Solutions
Share
Step: 1
(
f
+
g
)(
x
) =
f
(
x
) +
g
(
x
)
Step: 2
= 11
x
+ 3 + 11
x
+ 7
Step: 3
= 22
x
+ 10
Correct Answer is :
22
x
+ 10
Q10
If
f
(
x
) = 12
x
- 9
and
g
(
x
) = 12
x
+ 21,
find
(
f
+
g
)(
x
).
A
. 12
x
+ 12
B
. 24
x
+ 12
C
. 6
x
- 12
D
.
x
+ 12
Solutions
Share
Step: 1
(
f
+
g
)(
x
) =
f
(
x
) +
g
(
x
)
Step: 2
= 12
x
- 9 + 12
x
+ 21
Step: 3
= 24
x
+ 12
Correct Answer is :
24
x
+ 12
Q11
If
f
(
x
) = 2
x
,
g
(
x
) =
x
6
+ 8,
what is the value of
(
f
×
g
)(
x
)?
A
. 2
x
7
+ 16
x
B
.
x
7
+ 2
x
C
. 2
x
6
+ 16
x
D
. 2
x
8
+ 16
x
Solutions
Share
Step: 1
Given,
f
(
x
) = 2
x
and
g
(
x
) =
x
6
+ 8.
Step: 2
(
f
×
g
)(
x
) =
f
(
x
) ×
g
(
x
)
Step: 3
= 2
x
× (
x
6
+ 8) = 2
x
7
+ 16
x
Correct Answer is :
2
x
7
+ 16
x
Q12
If
f
(
x
) = 9
x
,
g
(
x
) =
x
5
,
what is the value of
(
f
g
)(
x
)?
A
. 9
x
B
.
x
5
C
. 9
x
6
D
.
9
x
4
Solutions
Share
Step: 1
Given,
f
(
x
) = 9
x
and
g
(
x
) =
x
5
Step: 2
(
f
g
)(
x
) =
f
(
x
)
g
(
x
)
,
g
(
x
) ≠ 0
Step: 3
=
9
x
x
5
Step: 4
=
9
x
4
Correct Answer is :
9
x
4
Q13
If
f
(
x
) = 7
x
2
,
g
(
x
) =
x
3
+ 8,
what is the value of
(
f
×
g
)(
x
)?
A
. 7
x
2
+ 56
x
B
. 7
x
6
+ 56
x
2
C
. 7
x
5
+ 8
D
. 7
x
5
+ 56
x
2
Solutions
Share
Step: 1
Given,
f
(
x
) = 7
x
2
and
g
(
x
) =
x
3
+ 8.
Step: 2
(
f
×
g
)(
x
) =
f
(
x
) ×
g
(
x
)
Step: 3
= 7
x
2
× (
x
3
+ 8) = 7
x
5
+ 56
x
2
Correct Answer is :
7
x
5
+ 56
x
2
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