3.3 Differentiating Inverse Functions

    Cards (20)

    • What does an inverse function do to the input and output of the original function?
      Reverses their roles
    • The relationship between a function f(x)f(x) and its inverse f−1(x)f^{ - 1}(x) is given by f^{ - 1}(f(x)) = x</latex> and <latex>f(f^{ - 1}(x)) = x
    • If f(x)=f(x) =2x 2x, then f−1(x)=f^{ - 1}(x) =x2 \frac{x}{2}.
    • What is the property of inverse functions when applied sequentially?
      Returns the original input
    • The relationship between a function f(x)f(x) and its inverse f−1(x)f^{ - 1}(x) is given by f^{ - 1}(f(x)) = x</latex> and <latex>f(f^{ - 1}(x)) = x
    • If f(x)=f(x) =2x 2x, then f−1(6)=f^{ - 1}(6) =3 3.
    • What does an inverse function reverse in the original function f(x)f(x)?

      Input and output
    • If f−1(f(x))=f^{ - 1}(f(x)) =x x, it means the inverse function undoes the original function.
    • f(f−1(x))=f(f^{ - 1}(x)) =x x demonstrates that the original function undoes the inverse
    • What is the notation for an inverse function?
      f−1(x)f^{ - 1}(x)
    • If f(a)=f(a) =b b, then f^{ - 1}(b) = a</latex> is true for all inverse functions.
    • What is the inverse function of f(x)=f(x) =2x 2x?

      f−1(x)=f^{ - 1}(x) =x2 \frac{x}{2}
    • The derivative of an inverse function at a point is the reciprocal of the derivative of the original function evaluated at the corresponding input to the inverse function.
    • If f(x)=f(x) =2x 2x, what is the value of f^{ - 1}(6)</latex>?

      33
    • Steps to verify the relationships between a function and its inverse
      1️⃣ Check if f−1(f(x))=f^{ - 1}(f(x)) =x x
      2️⃣ Check if f(f−1(x))=f(f^{ - 1}(x)) =x x
    • What is the relationship between f−1(x)f^{ - 1}(x) and f(x)f(x) in terms of composition?

      f−1(f(x))=f^{ - 1}(f(x)) =x x
    • If f(x) = 2x</latex>, then f−1(x)=f^{ - 1}(x) =x2 \frac{x}{2}. For x=x =3 3, f(3)=f(3) =6 6 and f−1(6)=f^{ - 1}(6) =3 3 demonstrates the input-output reversal property of inverse functions.
    • If f(x)=f(x) =x3 x^{3}, what is f′(x)f'(x)?

      3x23x^{2}
    • Steps to find the derivative of an inverse function (f−1)′(x)(f^{ - 1})'(x)

      1️⃣ Find the inverse function f−1(x)f^{ - 1}(x)
      2️⃣ Find the derivative f′(x)f'(x) of the original function
      3️⃣ Substitute f−1(x)f^{ - 1}(x) into f′(x)f'(x)
      4️⃣ Use the formula (f−1)′(x)=(f^{ - 1})'(x) =1f′(f−1(x)) \frac{1}{f'(f^{ - 1}(x))}
    • If f(x)=f(x) =2x+ 2x +3 3, what is f′(x)f'(x)?

      2