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Function Operations

• Operations on Functions: The primary operations you can perform on functions include addition, subtraction, multiplication, and division.

Addition of Functions

• Definition: Given two functions $f(x)$ and $g(x)$, the sum of these functions is denoted by $f + g$.

• Domain: The domain of the sum consists of the x-values that belong to the domains of both $f(x)$ and $g(x)$.

Subtraction of Functions

• Definition: The difference of two functions is denoted by $f - g$.

• Domain: Similar to addition, the domain is the intersection of the domains of both functions.

Multiplication of Functions

• Definition: The product of two functions is denoted by $f \times g$ or $f \backslash g$.

• Domain: The domain consists of x-values common to both functions' domains.

Division of Functions

• Definition: The quotient of functions is denoted as $\frac{f}{g}$.

• Domain: The domain of the quotient excludes x-values that make the denominator equal to zero.

Composition of Functions

• Definition: Composition of functions refers to a function being applied within another function. Denoted as $f(g(x))$.

• Order of Operations: Always start with the inside function and work your way out.

Inverse Functions

• Definition: If the point $(a, b)$ is on the function $f(x)$, then the point $(b, a)$ is on the inverse function $f^{-1}(x)$.

• Reflection: Every inverse function is a reflection over the line $y = x$.

Finding Inverses Algebraically

1. Replace $f(x)$ with $y$.

2. Swap $x$ and $y$.

3. Solve for $y$.

4. Replace $y$ with $f^{-1}(x)$ (the notation denoting the inverse).