Function Arithmetic (Sum, Difference, Product, Ratio):
\((f+g)(x) = f(x)+g(x)\) (Sum of functions)
\((f-g)(x) = f(x)-g(x)\) (Difference of functions)
\((fg)(x) = f(x)g(x)\) (Product of functions)
\((\frac{f}{g})(x) = \frac{f(x)}{g(x)}\), provided \(g(x) \neq 0\) (Quotient or Ratio of functions
Difference Quotient: For a given function \(f\), the expression \(\frac{f(x+h)-f(x)}{h}\) (or \(\frac{f(a+h)-f(a)}{h}\)).
Composition of Functions: The process of combining functions such that the output of one function becomes the input of another. The resulting function is known as a composite function.
Composite Function (Notation): For functions \(f\) and \(g\), the composite function, denoted \(f \circ g\), is defined by \((f \circ g)(x) = f(g(x))\).
Composition Operator (\(\circ\)): The open circle symbol used to denote function composition, as in \(f \circ g\).
Theorems
Domain of a Composite Function: The domain of the composite function \(f \circ g\) is all \(x\) such that \(x\) is in the domain of \(g\) and \(g(x)\) is in the domain of \(f\).
Common Mistakes
Incorrectly Using Grouping Symbols in Difference Quotients: When calculating \(f(x+h)-f(x)\), failing to use parentheses around \(f(x)\) if \(f(x)\) consists of multiple terms will lead to errors in subtraction. For example, for \(f(x) = x^2-x-2\), writing \(f(x+h) - x^2-x-2\) instead of \(f(x+h) - (x^2-x-2)\).
Confusing Function Composition with Function Multiplication: The product of functions \((fg)(x) = f(x)g(x)\) is not the same as the function composition \((f \circ g)(x) = f(g(x))\). In general, \(f(x)g(x) \neq f(g(x))\).