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# 3.5 inverse functions

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## 3.5 inverse functionsPresentation Transcript

• Inverse Functions
• Inverse Functions
A function f(x) = y takes an input x and produces one output y.
• Inverse Functions
A function f(x) = y takes an input x and produces one output y. Often we represent a function by the following figure.
f
y=f(x)
x
domian
range
View slide
• Inverse Functions
A function f(x) = y takes an input x and produces one output y. Often we represent a function by the following figure.
f
y=f(x)
x
domian
range
We like to reverse the operation, i.e., if we know the output y, what was (were) the input x?
View slide
• Inverse Functions
A function f(x) = y takes an input x and produces one output y. Often we represent a function by the following figure.
f
y=f(x)
x
domian
range
We like to reverse the operation, i.e., if we know the output y, what was (were) the input x?
This procedure of associating the output y to the input x may or may not be a function.
• Inverse Functions
A function f(x) = y takes an input x and produces one output y. Often we represent a function by the following figure.
f
y=f(x)
x
domian
range
We like to reverse the operation, i.e., if we know the output y, what was (were) the input x?
This procedure of associating the output y to the input x may or may not be a function.
If it is a function, it is called the inverse function of f(x) and it is denoted as f -1(y).
• Inverse Functions
A function f(x) = y takes an input x and produces one output y. Often we represent a function by the following figure.
f
y=f(x)
x
domian
range
We like to reverse the operation, i.e., if we know the output y, what was (were) the input x?
This procedure of associating the output y to the input x may or may not be a function.
If it is a function, it is called the inverse function of f(x) and it is denoted as f -1(y).
f
y=f(x)
x
• Inverse Functions
A function f(x) = y takes an input x and produces one output y. Often we represent a function by the following figure.
f
y=f(x)
x
domian
range
We like to reverse the operation, i.e., if we know the output y, what was (were) the input x?
This procedure of associating the output y to the input x may or may not be a function.
If it is a function, it is called the inverse function of f(x) and it is denoted as f -1(y).
f
y=f(x)
x=f-1(y)
f -1
• Inverse Functions
A function f(x) = y takes an input x and produces one output y. Often we represent a function by the following figure.
f
y=f(x)
x
domian
range
We like to reverse the operation, i.e., if we know the output y, what was (were) the input x?
This procedure of associating the output y to the input x may or may not be a function.
If it is a function, it is called the inverse function of f(x) and it is denoted as f -1(y).We say f(x) and f -1(y) are the inverse of each other.
f
y=f(x)
x=f-1(y)
f -1
• Inverse Functions
Example A:
a. The function y = f(x) = 2x takes the input x and double it to get the output y.
• Inverse Functions
Example A:
a. The function y = f(x) = 2x takes the input x and double it to get the output y. To reverse the operation, take an output y,
• Inverse Functions
Example A:
a. The function y = f(x) = 2x takes the input x and double it to get the output y. To reverse the operation, take an output y, divided it by 2 and we get back to the x.
• Inverse Functions
Example A:
a. The function y = f(x) = 2x takes the input x and double it to get the output y. To reverse the operation, take an output y, divided it by 2 and we get back to the x. In other words f -1(y) = y/2.
• Inverse Functions
Example A:
a. The function y = f(x) = 2x takes the input x and double it to get the output y. To reverse the operation, take an output y, divided it by 2 and we get back to the x. In other words f -1(y) = y/2.
So, for example, f -1(6) = 3 because f(3) = 6.
• Inverse Functions
Example A:
a. The function y = f(x) = 2x takes the input x and double it to get the output y. To reverse the operation, take an output y, divided it by 2 and we get back to the x. In other words f -1(y) = y/2.
So, for example, f -1(6) = 3 becausef(3) = 6.
b. Given y = f(x) = x2 and y = 9,
• Inverse Functions
Example A:
a. The function y = f(x) = 2x takes the input x and double it to get the output y. To reverse the operation, take an output y, divided it by 2 and we get back to the x. In other words f -1(y) = y/2.
So, for example, f -1(6) = 3 becausef(3) = 6.
b. Given y = f(x) = x2 and y = 9, there are two numbers, namely x = 3 and x = -3, associated to 9.
• Inverse Functions
Example A:
a. The function y = f(x) = 2x takes the input x and double it to get the output y. To reverse the operation, take an output y, divided it by 2 and we get back to the x. In other words f -1(y) = y/2.
So, for example, f -1(6) = 3 becausef(3) = 6.
b. Given y = f(x) = x2 and y = 9, there are two numbers, namely x = 3 and x = -3, associated to 9. Therefore, the reverse procedure is not a function.
• Inverse Functions
Example A:
a. The function y = f(x) = 2x takes the input x and double it to get the output y. To reverse the operation, take an output y, divided it by 2 and we get back to the x. In other words f -1(y) = y/2.
So, for example, f -1(6) = 3 becausef(3) = 6.
b. Given y = f(x) = x2 and y = 9, there are two numbers, namely x = 3 and x = -3, associated to 9. Therefore, the reverse procedure is not a function.
f(x)=x2
x=3
x=-3
y=9
not a function
• Inverse Functions
A function is one-to-one if different inputs produce different outputs.
• Inverse Functions
A function is one-to-one if different inputs produce different outputs. That is, f(x) is said to be
one-to-one if for every two inputs u and v such that
u  v, then f(u)  f(v).
• Inverse Functions
A function is one-to-one if different inputs produce different outputs. That is, f(x) is said to be
one-to-one if for every two inputs u and v such that
u  v, then f(u)  f(v).
u
f(u)
v
f(v)
u = v
f(u) = f(v)
a one-to-one function
• Inverse Functions
A function is one-to-one if different inputs produce different outputs. That is, f(x) is said to be
one-to-one if for every two inputs u and v such that
u  v, then f(u)  f(v).
u
u
f(u)
f(u)=f(v)
v
v
f(v)
u = v
u = v
f(u) = f(v)
not a one-to-one function
a one-to-one function
• Inverse Functions
A function is one-to-one if different inputs produce different outputs. That is, f(x) is said to be
one-to-one if for every two inputs u and v such that
u  v, then f(u)  f(v).
u
u
f(u)
f(u)=f(v)
v
v
f(v)
u = v
u = v
f(u) = f(v)
not a one-to-one function
a one-to-one function
Example B:
a. g(x) = 2x is one-to-one
• Inverse Functions
A function is one-to-one if different inputs produce different outputs. That is, f(x) is said to be
one-to-one if for every two inputs u and v such that
u  v, then f(u)  f(v).
u
u
f(u)
f(u)=f(v)
v
v
f(v)
u = v
u = v
f(u) = f(v)
not a one-to-one function
a one-to-one function
Example B:
a. g(x) = 2x is one-to-one
because if u  v, then 2u  2v.
• Inverse Functions
A function is one-to-one if different inputs produce different outputs. That is, f(x) is said to be
one-to-one if for every two inputs u and v such that
u  v, then f(u)  f(v).
u
u
f(u)
f(u)=f(v)
v
v
f(v)
u = v
u = v
f(u) = f(v)
not a one-to-one function
a one-to-one function
Example B:
a. g(x) = 2x is one-to-one
because if u  v, then 2u  2v.
b. f(x) = x2 is not one-to-one because 3  -3,
but f(3) = f(-3) = 9.
• Inverse Functions
Fact: If y = f(x) is one-to-one, then the reverse procedure for f(x) is a function
• Inverse Functions
Fact: If y = f(x) is one-to-one, then the reverse procedure for f(x) is a function i.e. f -1(y) exists.
Given y = f(x), to find f -1(y), just solve the equation
y = f(x) for x in terms of y.
• Inverse Functions
Fact: If y = f(x) is one-to-one, then the reverse procedure for f(x) is a function i.e. f -1(y) exists.
Given y = f(x), to find f -1(y), just solve the equation
y = f(x) for x in terms of y.
Example C:
Find the inverse function of y = f(x) = x – 5
3
4
• Inverse Functions
Fact: If y = f(x) is one-to-one, then the reverse procedure for f(x) is a function i.e. f -1(y) exists.
Given y = f(x), to find f -1(y), just solve the equation
y = f(x) for x in terms of y.
Example C:
Find the inverse function of y = f(x) = x – 5
Given y = x – 5 and solve for x.
3
4
3
4
• Inverse Functions
Fact: If y = f(x) is one-to-one, then the reverse procedure for f(x) is a function i.e. f -1(y) exists.
Given y = f(x), to find f -1(y), just solve the equation
y = f(x) for x in terms of y.
Example C:
Find the inverse function of y = f(x) = x – 5
Given y = x – 5 and solve for x.
Clear denominator: 4y = 3x – 20
3
4
3
4
• Inverse Functions
Fact: If y = f(x) is one-to-one, then the reverse procedure for f(x) is a function i.e. f -1(y) exists.
Given y = f(x), to find f -1(y), just solve the equation
y = f(x) for x in terms of y.
Example C:
Find the inverse function of y = f(x) = x – 5
Given y = x – 5 and solve for x.
Clear denominator: 4y = 3x – 20
4y + 20 = 3x
3
4
3
4
• Inverse Functions
Fact: If y = f(x) is one-to-one, then the reverse procedure for f(x) is a function i.e. f -1(y) exists.
Given y = f(x), to find f -1(y), just solve the equation
y = f(x) for x in terms of y.
Example C:
Find the inverse function of y = f(x) = x – 5
Given y = x – 5 and solve for x.
Clear denominator: 4y = 3x – 20
4y + 20 = 3x
x =
3
4
3
4
4y + 20
3
• Inverse Functions
Fact: If y = f(x) is one-to-one, then the reverse procedure for f(x) is a function i.e. f -1(y) exists.
Given y = f(x), to find f -1(y), just solve the equation
y = f(x) for x in terms of y.
Example C:
Find the inverse function of y = f(x) = x – 5
Given y = x – 5 and solve for x.
Clear denominator: 4y = 3x – 20
4y + 20 = 3x
x =
3
4
3
4
4y + 20
3
4y + 20
Hence f -1(y) =
3
• Inverse Functions
Reminder: If f(x) and f -1(y) are the inverse of each other, then f(a) = b if and only if a = f -1(b)
• Inverse Functions
Reminder: If f(x) and f -1(y) are the inverse of each other, then f(a) = b if and only if a = f -1(b)
Since we usually use x as the input variable for functions, we often use x instead of y as the variable for the inverse functions.
• Inverse Functions
Reminder: If f(x) and f -1(y) are the inverse of each other, then f(a) = b if and only if a = f -1(b)
Since we usually use x as the input variable for functions, we often use x instead of y as the variable for the inverse functions. Hence in example C, the
4x + 20
f -1(x) =
.
3
• Inverse Functions
Reminder: If f(x) and f -1(y) are the inverse of each other, then f(a) = b if and only if a = f -1(b)
Since we usually use x as the input variable for functions, we often use x instead of y as the variable for the inverse functions. Hence in example C, the
4x + 20
f -1(x) =
.
3
Fact: If f(x) and f -1(y) are the inverse of each other, then f -1(f(x)) = x and f(f -1(x)) = x.
• Inverse Functions
Reminder: If f(x) and f -1(y) are the inverse of each other, then f(a) = b if and only if a = f -1(b)
Since we usually use x as the input variable for functions, we often use x instead of y as the variable for the inverse functions. Hence in example C, the
4x + 20
f -1(x) =
.
3
Fact: If f(x) and f -1(y) are the inverse of each other, then f -1(f(x)) = x and f(f -1(x)) = x.
f
f(x)
x
• Inverse Functions
Reminder: If f(x) and f -1(y) are the inverse of each other, then f(a) = b if and only if a = f -1(b)
Since we usually use x as the input variable for functions, we often use x instead of y as the variable for the inverse functions. Hence in example C, the
4x + 20
f -1(x) =
.
3
Fact: If f(x) and f -1(y) are the inverse of each other, then f -1(f(x)) = x and f(f -1(x)) = x.
f
f(x)
x
f -1
f -1(f(x)) = x
• Inverse Functions
Reminder: If f(x) and f -1(y) are the inverse of each other, then f(a) = b if and only if a = f -1(b)
Since we usually use x as the input variable for functions, we often use x instead of y as the variable for the inverse functions. Hence in example C, the
4x + 20
f -1(x) =
.
3
Fact: If f(x) and f -1(y) are the inverse of each other, then f -1(f(x)) = x and f(f -1(x)) = x.
f
x
f(x)
f-1(x)
x
f -1
f -1
f -1(f(x)) = x
• Inverse Functions
Reminder: If f(x) and f -1(y) are the inverse of each other, then f(a) = b if and only if a = f -1(b)
Since we usually use x as the input variable for functions, we often use x instead of y as the variable for the inverse functions. Hence in example C, the
4x + 20
f -1(x) =
.
3
Fact: If f(x) and f -1(y) are the inverse of each other, then f -1(f(x)) = x and f(f -1(x)) = x.
f
f
x
f(x)
f-1(x)
x
f -1
f -1
f -1(f(x)) = x
f(f -1(x)) = x
• Inverse Functions
Example D:
2x – 3
a. Given f(x) = find f -1(x).
,
x + 2
• Inverse Functions
Example D:
2x – 3
a. Given f(x) = find f -1(x).
,
x + 2
2x – 3
Set y = and solve for x in term of y.
,
x + 2
• Inverse Functions
Example D:
2x – 3
a. Given f(x) = find f -1(x).
,
x + 2
2x – 3
Set y = and solve for x in term of y.
,
x + 2
Clear the denominator, we get
y(x + 2) = 2x – 3
• Inverse Functions
Example D:
2x – 3
a. Given f(x) = find f -1(x).
,
x + 2
2x – 3
Set y = and solve for x in term of y.
,
x + 2
Clear the denominator, we get
y(x + 2) = 2x – 3
yx + 2y = 2x – 3
collect and isolate x
• Inverse Functions
Example D:
2x – 3
a. Given f(x) = find f -1(x).
,
x + 2
2x – 3
Set y = and solve for x in term of y.
,
x + 2
Clear the denominator, we get
y(x + 2) = 2x – 3
yx + 2y = 2x – 3
collect and isolate x
yx – 2x = –2y – 3
• Inverse Functions
Example D:
2x – 3
a. Given f(x) = find f -1(x).
,
x + 2
2x – 3
Set y = and solve for x in term of y.
,
x + 2
Clear the denominator, we get
y(x + 2) = 2x – 3
yx + 2y = 2x – 3
collect and isolate x
yx – 2x = –2y – 3
(y – 2)x = –2y – 3
• Inverse Functions
Example D:
2x – 3
a. Given f(x) = find f -1(x).
,
x + 2
2x – 3
Set y = and solve for x in term of y.
,
x + 2
Clear the denominator, we get
y(x + 2) = 2x – 3
yx + 2y = 2x – 3
collect and isolate x
yx – 2x = –2y – 3
(y – 2)x = –2y – 3
–2y – 3
–2y – 3
Hence f -1(y) =
x =
y – 2
y – 2
• Inverse Functions
Example D:
2x – 3
a. Given f(x) = find f -1(x).
,
x + 2
2x – 3
Set y = and solve for x in term of y.
,
x + 2
Clear the denominator, we get
y(x + 2) = 2x – 3
yx + 2y = 2x – 3
collect and isolate x
yx – 2x = –2y – 3
(y – 2)x = –2y – 3
–2y – 3
–2y – 3
Hence f -1(y) =
x =
y – 2
y – 2
Write the answer using x as the variable:
–2x – 3
f -1(x) =
x – 2
• Inverse Functions
b. Verify that f(f -1(x)) = x
• Inverse Functions
b. Verify that f(f -1(x)) = x
2x – 3
–2x – 3
We've f(x) = and
,
f -1(x) =
x + 2
x – 2
• Inverse Functions
b. Verify that f(f -1(x)) = x
2x – 3
–2x – 3
We've f(x) = and
,
f -1(x) =
x + 2
x – 2
–2x – 3
f(f -1(x)) = f( )
x – 2
• Inverse Functions
b. Verify that f(f -1(x)) = x
2x – 3
–2x – 3
We've f(x) = and
,
f -1(x) =
x + 2
x – 2
–2x – 3
f(f -1(x)) = f( )
x – 2
–2x – 3
(
)
– 3
2
x – 2
=
–2x – 3
(
)
+ 2
x – 2
• Inverse Functions
b. Verify that f(f -1(x)) = x
2x – 3
–2x – 3
We've f(x) = and
,
f -1(x) =
x + 2
x – 2
–2x – 3
f(f -1(x)) = f( )
x – 2
–2x – 3
[
]
(
)
– 3
(x – 2)
2
x – 2
=
–2x – 3
[
]
(
)
(x – 2)
+ 2
x – 2
• Inverse Functions
b. Verify that f(f -1(x)) = x
2x – 3
–2x – 3
We've f(x) = and
,
f -1(x) =
x + 2
x – 2
–2x – 3
f(f -1(x)) = f( )
x – 2
–2x – 3
[
]
(
)
– 3
(x – 2)
2
x – 2
=
–2x – 3
[
]
(
)
(x – 2)
+ 2
x – 2
2(-2x – 3) – 3(x – 2)
=
(-2x – 3) + 2(x – 2)
• Inverse Functions
b. Verify that f(f -1(x)) = x
2x – 3
–2x – 3
We've f(x) = and
,
f -1(x) =
x + 2
x – 2
–2x – 3
f(f -1(x)) = f( )
x – 2
–2x – 3
[
]
(
)
– 3
(x – 2)
2
x – 2
=
–2x – 3
[
]
(
)
(x – 2)
+ 2
x – 2
2(-2x – 3) – 3(x – 2)
=
(-2x – 3) + 2(x – 2)
-4x – 6 – 3x + 6
=
-2x – 3 + 2x – 4
• Inverse Functions
b. Verify that f(f -1(x)) = x
2x – 3
–2x – 3
We've f(x) = and
,
f -1(x) =
x + 2
x – 2
–2x – 3
f(f -1(x)) = f( )
x – 2
–2x – 3
[
]
(
)
– 3
(x – 2)
2
x – 2
=
–2x – 3
[
]
(
)
(x – 2)
+ 2
x – 2
2(-2x – 3) – 3(x – 2)
=
(-2x – 3) + 2(x – 2)
-4x – 6 – 3x + 6
-7x
=
= x
=
-7
-2x – 3 + 2x – 4
HW. Verify that f -1(f(x)) = x