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3.5 inverse functions 3.5 inverse functions Presentation 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
    answer may be written as
    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
    answer may be written as
    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
    answer may be written as
    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
    answer may be written as
    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
    answer may be written as
    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
    answer may be written as
    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