Functions: Inverse and Composite

Study Notes

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.

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.

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.

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.

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.

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.