Functions: Inverse and Composite

Inverse Functions

• A function f has an inverse function f^(-1) if and only if f is one-to-one (injective).
• The inverse function switches the input and output values of the original function.
• The domain of f becomes the range of f^(-1), and vice versa.
• To find the inverse function, swap the x and y variables in the function equation and then solve for y.

Composite Functions

• A composite function is a function of a function, denoted as (f ∘ g)(x) = f(g(x)).
• The inner function g is evaluated first, and then the outer function f is applied to the result.
• Composite functions can be evaluated by substituting the inner function into the outer function.
• Chain rule is used to differentiate composite functions.

Limits

• A limit represents the behavior of a function as the input (x) approaches a specific value.
• Notation: lim x→a f(x) = L means the limit of f(x) as x approaches a is L.
• Properties of limits:
  • Linearity: lim x→a [af(x) + bg(x)] = a lim x→a f(x) + b lim x→a g(x)
  • Homogeneity: lim x→a [f(x)g(x)] = (lim x→a f(x))(lim x→a g(x))
  • Sum: lim x→a [f(x) + g(x)] = lim x→a f(x) + lim x→a g(x)
• Types of limits:
  • One-sided limits: lim x→a⁺ f(x) and lim x→a⁻ f(x), where a⁺ and a⁻ denote approaching from the right and left, respectively.
  • Infinite limits: lim x→a f(x) = ±∞, indicating the function grows without bound as x approaches a.

Inverse Functions

• A function is one-to-one (injective) if and only if it has an inverse function.
• The inverse function reverses the input and output values of the original function.
• The domain of the original function becomes the range of the inverse function, and vice versa.
• To find the inverse function, swap the x and y variables in the function equation and then solve for y.

Composite Functions

• A composite function is a function of a function, denoted as (f ∘ g)(x) = f(g(x)).
• The inner function is evaluated first, followed by the outer function.
• Composite functions can be evaluated by substituting the inner function into the outer function.
• The chain rule is used to differentiate composite functions.

Limits

• A limit represents the behavior of a function as the input approaches a specific value.
• The notation lim x→a f(x) = L means the limit of f(x) as x approaches a is L.
• Properties of limits include:
  • Linearity: lim x→a [af(x) + bg(x)] = a lim x→a f(x) + b lim x→a g(x)
  • Homogeneity: lim x→a [f(x)g(x)] = (lim x→a f(x))(lim x→a g(x))
  • Sum: lim x→a [f(x) + g(x)] = lim x→a f(x) + lim x→a g(x)
• Types of limits include:
  • One-sided limits: lim x→a⁺ f(x) and lim x→a⁻ f(x), where a⁺ and a⁻ denote approaching from the right and left, respectively.
  • Infinite limits: lim x→a f(x) = ±∞, indicating the function grows without bound as x approaches a.