Composite Functions
Composite functions are formed by applying one function after another. This operation is denoted by the symbol "∘" and is read as "composed with" or "after".
For two functions $f(x)$ and $g(x)$, their composition $(f \circ g)(x)$ is defined as:
$$(f \circ g)(x) = f(g(x))$$
This means we first apply function $g$ to $x$, and then apply function $f$ to the result.
ExampleLet $f(x) = x^2$ and $g(x) = x + 1$. Then:
$(f \circ g)(x) = f(g(x)) = f(x + 1) = (x + 1)^2 = x^2 + 2x + 1$
$(g \circ f)(x) = g(f(x)) = g(x^2) = x^2 + 1$
Note that $(f \circ g)(x) \neq (g \circ f)(x)$ in general.
NoteThe order of composition matters. Changing the order often results in a different function.
Identity Function
The identity function, denoted as $I(x)$ or $id(x)$, is a function that returns its input unchanged:
$$I(x) = x$$
This function plays a crucial role in function theory, similar to how the number 1 is important in multiplication.
ExampleFor any function $f(x)$: $(f \circ I)(x) = f(I(x)) = f(x)$ $(I \circ f)(x) = I(f(x)) = f(x)$
Inverse Functions
For a function $f(x)$, its inverse function, denoted as $f^{-1}(x)$, "undoes" what $f$ does. Formally:
$$(f \circ f^{-1})(x) = (f^{-1} \circ f)(x) = x$$
To find the inverse of a function:
- Replace $f(x)$ with $y$
- Swap $x$ and $y$
