Section 1-3Function Notations
Warm-up1. If , find y when x = 0.
y = 3x − 2 2. If ,find y when x = 3.
y = 4x + x2
16
3. If ,find y when x = 5.
y =4 − xx2
4. If ,find y when x = 6.
y = 3x −1
Warm-up1. If , find y when x = 0.
y = 3x − 2 2. If ,find y when x = 3.
y = 4x + x2
16
3. If ,find y when x = 5.
y =4 − xx2
4. If ,find y when x = 6.
y = 3x −1
y = 3(0) − 2
Warm-up1. If , find y when x = 0.
y = 3x − 2 2. If ,find y when x = 3.
y = 4x + x2
16
3. If ,find y when x = 5.
y =4 − xx2
4. If ,find y when x = 6.
y = 3x −1
y = 3(0) − 2 y = 0 − 2
Warm-up1. If , find y when x = 0.
y = 3x − 2 2. If ,find y when x = 3.
y = 4x + x2
16
3. If ,find y when x = 5.
y =4 − xx2
4. If ,find y when x = 6.
y = 3x −1
y = 3(0) − 2 y = 0 − 2
y = −2
Warm-up1. If , find y when x = 0.
y = 3x − 2 2. If ,find y when x = 3.
y = 4x + x2
16
3. If ,find y when x = 5.
y =4 − xx2
4. If ,find y when x = 6.
y = 3x −1
y = 3(0) − 2 y = 0 − 2
y = −2
y = 4(3) + 32
16
Warm-up1. If , find y when x = 0.
y = 3x − 2 2. If ,find y when x = 3.
y = 4x + x2
16
3. If ,find y when x = 5.
y =4 − xx2
4. If ,find y when x = 6.
y = 3x −1
y = 3(0) − 2 y = 0 − 2
y = −2
y = 4(3) + 32
16 y = 12 + 916
Warm-up1. If , find y when x = 0.
y = 3x − 2 2. If ,find y when x = 3.
y = 4x + x2
16
3. If ,find y when x = 5.
y =4 − xx2
4. If ,find y when x = 6.
y = 3x −1
y = 3(0) − 2 y = 0 − 2
y = −2
y = 4(3) + 32
16 y = 12 + 916
y = 12 916
Warm-up1. If , find y when x = 0.
y = 3x − 2 2. If ,find y when x = 3.
y = 4x + x2
16
3. If ,find y when x = 5.
y =4 − xx2
4. If ,find y when x = 6.
y = 3x −1
y = 3(0) − 2 y = 0 − 2
y = −2
y = 4(3) + 32
16 y = 12 + 916
y = 12 916 =
20116
Warm-up1. If , find y when x = 0.
y = 3x − 2 2. If ,find y when x = 3.
y = 4x + x2
16
3. If ,find y when x = 5.
y =4 − xx2
4. If ,find y when x = 6.
y = 3x −1
y = 3(0) − 2 y = 0 − 2
y = −2
y = 4(3) + 32
16 y = 12 + 916
y = 12 916 =
20116
y =
4 − 552
Warm-up1. If , find y when x = 0.
y = 3x − 2 2. If ,find y when x = 3.
y = 4x + x2
16
3. If ,find y when x = 5.
y =4 − xx2
4. If ,find y when x = 6.
y = 3x −1
y = 3(0) − 2 y = 0 − 2
y = −2
y = 4(3) + 32
16 y = 12 + 916
y = 12 916 =
20116
y =
4 − 552
y =−125
Warm-up1. If , find y when x = 0.
y = 3x − 2 2. If ,find y when x = 3.
y = 4x + x2
16
3. If ,find y when x = 5.
y =4 − xx2
4. If ,find y when x = 6.
y = 3x −1
y = 3(0) − 2 y = 0 − 2
y = −2
y = 4(3) + 32
16 y = 12 + 916
y = 12 916 =
20116
y =
4 − 552
y =−125
y = 36−1
Warm-up1. If , find y when x = 0.
y = 3x − 2 2. If ,find y when x = 3.
y = 4x + x2
16
3. If ,find y when x = 5.
y =4 − xx2
4. If ,find y when x = 6.
y = 3x −1
y = 3(0) − 2 y = 0 − 2
y = −2
y = 4(3) + 32
16 y = 12 + 916
y = 12 916 =
20116
y =
4 − 552
y =−125
y = 36−1 y = 35
Warm-up1. If , find y when x = 0.
y = 3x − 2 2. If ,find y when x = 3.
y = 4x + x2
16
3. If ,find y when x = 5.
y =4 − xx2
4. If ,find y when x = 6.
y = 3x −1
y = 3(0) − 2 y = 0 − 2
y = −2
y = 4(3) + 32
16 y = 12 + 916
y = 12 916 =
20116
y =
4 − 552
y =−125
y = 36−1 y = 35
y = 243
Euler Notation:
Euler Notation: f(x), read as “f of x”
Euler Notation: f(x), read as “f of x”The variable inside the parentheses is the independent variable, while the entire “f(x)” is the dependent variable. This means y=f(x).
Euler Notation: f(x), read as “f of x”The variable inside the parentheses is the independent variable, while the entire “f(x)” is the dependent variable. This means y=f(x).
Argument:
Euler Notation: f(x), read as “f of x”The variable inside the parentheses is the independent variable, while the entire “f(x)” is the dependent variable. This means y=f(x).
Argument: Yet another word to represent the independent variable; can be filled with anything that is in the possible domain
Euler Notation: f(x), read as “f of x”The variable inside the parentheses is the independent variable, while the entire “f(x)” is the dependent variable. This means y=f(x).
Argument: Yet another word to represent the independent variable; can be filled with anything that is in the possible domain
Value:
Euler Notation: f(x), read as “f of x”The variable inside the parentheses is the independent variable, while the entire “f(x)” is the dependent variable. This means y=f(x).
Argument: Yet another word to represent the independent variable; can be filled with anything that is in the possible domain
Value: The dependent variable
Example 1 P(x) = x3,Q(x) = 3x + 3,R(x) = x
2x − 4Evaluate the following:
a. P(4) c. R(2) b. Q(−3)
Example 1 P(x) = x3,Q(x) = 3x + 3,R(x) = x
2x − 4Evaluate the following:
a. P(4) c. R(2)
P(4) = 43 b. Q(−3)
Example 1 P(x) = x3,Q(x) = 3x + 3,R(x) = x
2x − 4Evaluate the following:
a. P(4) c. R(2)
P(4) = 43
P(4) = 64
b. Q(−3)
Example 1 P(x) = x3,Q(x) = 3x + 3,R(x) = x
2x − 4Evaluate the following:
a. P(4) c. R(2)
P(4) = 43
P(4) = 64 Q(−3) = 3(−3) + 3
b. Q(−3)
Example 1 P(x) = x3,Q(x) = 3x + 3,R(x) = x
2x − 4Evaluate the following:
a. P(4) c. R(2)
P(4) = 43
P(4) = 64 Q(−3) = −9 + 3 Q(−3) = 3(−3) + 3
b. Q(−3)
Example 1 P(x) = x3,Q(x) = 3x + 3,R(x) = x
2x − 4Evaluate the following:
a. P(4) c. R(2)
P(4) = 43
P(4) = 64
Q(−3) = −6 Q(−3) = −9 + 3
Q(−3) = 3(−3) + 3 b. Q(−3)
Example 1 P(x) = x3,Q(x) = 3x + 3,R(x) = x
2x − 4Evaluate the following:
a. P(4) c. R(2)
P(4) = 43
P(4) = 64
Q(−3) = −6 Q(−3) = −9 + 3
Q(−3) = 3(−3) + 3 b. Q(−3)
R(2) = 2
2(2) − 4
Example 1 P(x) = x3,Q(x) = 3x + 3,R(x) = x
2x − 4Evaluate the following:
a. P(4) c. R(2)
P(4) = 43
P(4) = 64
Q(−3) = −6 Q(−3) = −9 + 3
Q(−3) = 3(−3) + 3 b. Q(−3)
R(2) = 2
2(2) − 4
R(2) = 2
4 − 4
Example 1 P(x) = x3,Q(x) = 3x + 3,R(x) = x
2x − 4Evaluate the following:
a. P(4) c. R(2)
P(4) = 43
P(4) = 64
Q(−3) = −6 Q(−3) = −9 + 3
Q(−3) = 3(−3) + 3 b. Q(−3)
R(2) = 2
2(2) − 4
R(2) = 2
4 − 4
R(2) = 2
0
Example 1 P(x) = x3,Q(x) = 3x + 3,R(x) = x
2x − 4Evaluate the following:
a. P(4) c. R(2)
P(4) = 43
P(4) = 64
Q(−3) = −6 Q(−3) = −9 + 3
Q(−3) = 3(−3) + 3 b. Q(−3)
R(2) = 2
2(2) − 4
R(2) = 2
4 − 4
R(2) = 2
0Undefined! 2 is not in the domain!
Example 1 d. P(−2) e. Q( 1
3 ) f. R(5)
Example 1 d. P(−2) e. Q( 1
3 ) f. R(5)
P(−2) = −8
Example 1 d. P(−2) e. Q( 1
3 ) f. R(5)
P(−2) = −8 Q( 13 ) = 4
Example 1 d. P(−2) e. Q( 1
3 ) f. R(5)
P(−2) = −8 Q( 13 ) = 4 R(5) = 5
6
Mapping Notation
Mapping NotationAlso known as Arrow Notation
Mapping NotationAlso known as Arrow Notation
A:x →
Mapping NotationAlso known as Arrow Notation
A:x →
Reads “A maps x onto”
Mapping NotationAlso known as Arrow Notation
A:x →
Reads “A maps x onto”
Still identifies the independent variable (after colon) and dependent variable (A:x)
Example 2Evaluate using the functions in Example 1.
a. P : 1 → b. Q : 2 → c. R : 3 →
Example 2Evaluate using the functions in Example 1.
a. P : 1 → b. Q : 2 → c. R : 3 →
P : 1 → 13
Example 2Evaluate using the functions in Example 1.
a. P : 1 → b. Q : 2 → c. R : 3 →
P : 1 → 13
= 1
Example 2Evaluate using the functions in Example 1.
a. P : 1 → b. Q : 2 → c. R : 3 →
P : 1 → 13
= 1
1
Example 2Evaluate using the functions in Example 1.
a. P : 1 → b. Q : 2 → c. R : 3 →
P : 1 → 13
= 1
1
Q : 2 → 3(2) + 3
Example 2Evaluate using the functions in Example 1.
a. P : 1 → b. Q : 2 → c. R : 3 →
P : 1 → 13
= 1
1
Q : 2 → 3(2) + 3
= 6 + 3
Example 2Evaluate using the functions in Example 1.
a. P : 1 → b. Q : 2 → c. R : 3 →
P : 1 → 13
= 1
1
Q : 2 → 3(2) + 3
= 6 + 3
= 9
Example 2Evaluate using the functions in Example 1.
a. P : 1 → b. Q : 2 → c. R : 3 →
P : 1 → 13
= 1
1
Q : 2 → 3(2) + 3
= 6 + 3
= 9
9
Example 2Evaluate using the functions in Example 1.
a. P : 1 → b. Q : 2 → c. R : 3 →
P : 1 → 13
= 1
1
Q : 2 → 3(2) + 3
= 6 + 3
= 9
9
R : 3 →
32(3) − 4
Example 2Evaluate using the functions in Example 1.
a. P : 1 → b. Q : 2 → c. R : 3 →
P : 1 → 13
= 1
1
Q : 2 → 3(2) + 3
= 6 + 3
= 9
9
R : 3 →
32(3) − 4
=
36 − 4
Example 2Evaluate using the functions in Example 1.
a. P : 1 → b. Q : 2 → c. R : 3 →
P : 1 → 13
= 1
1
Q : 2 → 3(2) + 3
= 6 + 3
= 9
9
R : 3 →
32(3) − 4
=
36 − 4
=
32
Example 2Evaluate using the functions in Example 1.
a. P : 1 → b. Q : 2 → c. R : 3 →
P : 1 → 13
= 1
1
Q : 2 → 3(2) + 3
= 6 + 3
= 9
9
R : 3 →
32(3) − 4
=
36 − 4
=
32
32
Example 3The area of a circle is a function of its radius.
Rewrite the formula using the following notations. A = πr2
a. Euler’s Notation b. Mapping Notation
Example 3The area of a circle is a function of its radius.
Rewrite the formula using the following notations. A = πr2
a. Euler’s Notation b. Mapping Notation
A(r ) = πr2
Example 3The area of a circle is a function of its radius.
Rewrite the formula using the following notations. A = πr2
a. Euler’s Notation b. Mapping Notation
A(r ) = πr2 A : r → πr2
Example 4If , then a : x → 2x + 7 a : 14 → ?
Example 4If , then a : x → 2x + 7 a : 14 → ?
a : 14 → 2(14) + 7
Example 4If , then a : x → 2x + 7 a : 14 → ?
a : 14 → 2(14) + 7 = 35
Example 4If , then a : x → 2x + 7 a : 14 → ?
a : 14 → 2(14) + 7 = 35
a : 14 → 35
Example 5Suppose . f (x) = 4x2 − 2x + 9
Find . f (−3)
Example 5Suppose . f (x) = 4x2 − 2x + 9
f (−3) = 4(−3)2 − 2(−3) + 9
Find . f (−3)
Example 5Suppose . f (x) = 4x2 − 2x + 9
f (−3) = 4(−3)2 − 2(−3) + 9
= 4(9) + 6 + 9
Find . f (−3)
Example 5Suppose . f (x) = 4x2 − 2x + 9
f (−3) = 4(−3)2 − 2(−3) + 9
= 4(9) + 6 + 9
= 36 + 6 + 9
Find . f (−3)
Example 5Suppose . f (x) = 4x2 − 2x + 9
f (−3) = 4(−3)2 − 2(−3) + 9
= 4(9) + 6 + 9
= 36 + 6 + 9
= 51
Find . f (−3)
Example 5Suppose . f (x) = 4x2 − 2x + 9
f (−3) = 4(−3)2 − 2(−3) + 9
= 4(9) + 6 + 9
= 36 + 6 + 9
= 51
f (−3) = 51
Find . f (−3)
Homework
Homework
p. 22 #1-26
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