Properties of Inverse Functions Quiz

Definition of Inverse Function

• A function f has an inverse function f^(-1) if and only if f is both one-to-one (injective) and onto (surjective).
• In other words, f must be a bijection (a one-to-one correspondence between the domain and range).

Properties of Inverse Functions

• The inverse of f is denoted by f^(-1) and is defined as:
  • f^(-1)(y) = x if and only if f(x) = y
• The domain of f^(-1) is the range of f, and the range of f^(-1) is the domain of f.
• f(f^(-1)(x)) = x and f^(-1)(f(x)) = x

Finding Inverse Functions

• To find the inverse of a function f, follow these steps:
  1. Write y = f(x)
  2. Interchange x and y to get x = f(y)
  3. Solve for y in terms of x to get y = f^(-1)(x)
• Example: Find the inverse of f(x) = 2x + 1
  1. Write y = 2x + 1
  2. Interchange x and y to get x = 2y + 1
  3. Solve for y to get y = (x - 1)/2 Thus, f^(-1)(x) = (x - 1)/2

Applications of Inverse Functions

• Inverse functions are used to solve equations and inequalities.
• They are also used in various fields such as physics, engineering, and economics to model real-world phenomena.
• Example: Find the input value of x that produces an output value of 5 for the function f(x) = 2x + 1.
  • Use the inverse function f^(-1)(x) = (x - 1)/2 to find x = f^(-1)(5) = (5 - 1)/2 = 2

Definition of Inverse Function

• A function f has an inverse f^(-1) if it is both one-to-one (injective) and onto (surjective).
• A bijection establishes a one-to-one correspondence between the function's domain and range.

Properties of Inverse Functions

• The inverse function notation is f^(-1) and defined by f^(-1)(y) = x if f(x) = y.
• The domain of the inverse f^(-1) corresponds to the range of f, while the range of f^(-1) reflects the domain of f.
• Fundamental property: f(f^(-1)(x)) = x and f^(-1)(f(x)) = x.

Finding Inverse Functions

• To derive the inverse of a function f, employ the following steps:
  • Begin with the equation y = f(x).
  • Swap x and y resulting in x = f(y).
  • Rearrange and solve for y, resulting in y = f^(-1)(x).
• Example process using f(x) = 2x + 1:
  • Set y = 2x + 1.
  • Interchange to get x = 2y + 1.
  • Solve to find y = (x - 1)/2, indicating f^(-1)(x) = (x - 1)/2.

Applications of Inverse Functions

• Used for solving equations and inequalities across various disciplines.
• Vital in fields like physics, engineering, and economics for modeling real-world scenarios.
• Example application: To find x producing f(x) = 5 in f(x) = 2x + 1, utilize the inverse function:
  • f^(-1)(5) = (5 - 1)/2 = 2.