3.3 Differentiating Inverse Functions

Cards (48)

If f(x) = 2x + 3, what is f⁻¹(x)?
(x - 3) / 2
If f(f⁻¹(x)) = x, then f(x) and f⁻¹(x) are inverses. 
True
Verify that f(x) = 2x + 3 and f⁻¹(x) = (x - 3) / 2 are inverses by evaluating f(f⁻¹(x)).
x
It's important not to confuse f⁻¹(x) with 1/f(x)
Verify that f(x) = 2x + 3 and f⁻¹(x) = (x - 3) / 2 are inverses by evaluating f(f⁻¹(x)).
x
The inverse function of f(x) = 2x + 3 is f⁻¹(x) = (x - 3) / 2
Applying an inverse function "undoes" the original function. 
True
What is the formula for the derivative of an inverse function (f⁻¹)'(x)?
$(f^{ - 1})'(x) =$ $\frac{1}{f'(f^{ - 1}(x))}$
The derivative of the inverse function for f(x) = 2x + 3 is (f⁻¹)'(x) = 1/2. 
True
The notation f⁻¹(x) does not represent the reciprocal of f(x), but its inverse.
If f(f⁻¹(x)) = x, then f(x) and f⁻¹(x) are inverses
What is the formula for the derivative of an inverse function (f⁻¹(x))' at x?
(f⁻¹)'(x) = 1 / f'(f⁻¹(x))
Given f(x) = 2x + 3 and f⁻¹(x) = (x - 3) / 2, what is (f⁻¹)'(x)?
1 / 2
What does the notation f⁻¹(x) represent for a function f(x)?
The inverse of f(x)
To verify if two functions are inverses, evaluate f(f⁻¹(x)) and ensure it equals x
To verify that two functions are inverses, evaluate f(f⁻¹(x)) and ensure it simplifies to x
Applying an inverse function should "undo" the original function. 
True
The -1 in the notation f⁻¹(x) indicates the reciprocal of f(x).
False
To verify if two functions are inverses, evaluate f⁻¹(f(x)) and ensure it equals x
To verify if two functions are inverses, we use the composition
The composition f⁻¹(f(x)) for f(x) = 2x + 3 equals x
The derivative of the inverse function is the reciprocal of the derivative of the original function evaluated at f⁻¹(x)
To find the derivative of an inverse function, we use the formula (f⁻¹)'(x) = 1 / f'(f⁻¹(x))
Applying an inverse function "undoes" the original function. 
True
Steps to verify if two functions f(x) and f⁻¹(x) are inverses using the composition property
1️⃣ Evaluate f(f⁻¹(x))
2️⃣ Check if f(f⁻¹(x)) = x
3️⃣ Evaluate f⁻¹(f(x))
4️⃣ Check if f⁻¹(f(x)) = x
The derivative of f⁻¹(x) is the reciprocal of the derivative of f(x) evaluated at f⁻¹(x)
(f⁻¹)'(x) = 1 / f'(f⁻¹(x)) is the formula for the derivative of an inverse function. 
True
The definition of the derivative is (f⁻¹)'(x) = lim_{h \to 0} \frac{f⁻¹(x + h) - f⁻¹(x)}{h}, which can be used to verify the derivative
The definition of the derivative states that (f⁻¹)'(x) = \lim_{h \to 0} \frac{f⁻¹(x + h) - f⁻¹(x)}{h}
What is the rate of change of the quantity demanded with respect to the price if f(x) = 2x + 3 models their relationship?
1/2
The expression f⁻¹(x) is the same as 1/f(x).
False
Steps to verify if two functions are inverses using composition:
1️⃣ Evaluate f(f⁻¹(x))
2️⃣ Evaluate f⁻¹(f(x))
3️⃣ Check if both equal x
Steps to verify if f(x) = 2x + 3 and f⁻¹(x) = (x - 3) / 2 are inverses:
1️⃣ Evaluate f(f⁻¹(x))
2️⃣ Evaluate f⁻¹(f(x))
3️⃣ Simplify both compositions
4️⃣ Check if both equal x
If f(x) = 2x + 3, what is f⁻¹(x)?
(x - 3) / 2
Steps to verify if two functions are inverses using composition:
1️⃣ Evaluate f(f⁻¹(x))
2️⃣ Evaluate f⁻¹(f(x))
3️⃣ Check if both equal x
Verify that f(x) = 2x + 3 and f⁻¹(x) = (x - 3) / 2 are inverses by evaluating f⁻¹(f(x)).
x
Steps to verify two functions are inverses
1️⃣ Evaluate f(f⁻¹(x))
2️⃣ If f(f⁻¹(x)) = x, continue
3️⃣ Evaluate f⁻¹(f(x))
4️⃣ If f⁻¹(f(x)) = x, the functions are inverses
Two functions are inverses if their composition equals x in both directions. 
True
What is the derivative of f(x) = 2x + 3?
2
The derivative of the inverse function for f(x) = 2x + 3 is a constant value of 1/2.
True