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 $y$
- Solve for $y$
- Replace $y$ with $f^{-1}(x)$
Find the inverse function for $f(x) = 2x + 3$
- $y = 2x + 3$
- $x = 2y + 3$
- $x - 3 = 2y$
- $\frac{x - 3}{2} = y$
- $f^{-1}(x) = \frac{x - 3}{2}$
Verify: $(f \circ f^{-1})(x) = f(\frac{x - 3}{2}) = 2(\frac{x - 3}{2}) + 3 = x - 3 + 3 = x$
Common MistakeNot all functions have inverses. Only one-to-one functions (injective functions) have inverses.
One-to-One Functions and Inverses
A function is one-to-one (injective) if each element in the codomain is paired with at most one element in the domain. Graphically, this means that any horizontal line intersects the graph of the function at most once.
TipTo check if a function is one-to-one, use the horizontal line test: if any horizontal line intersects the graph more than once, the function is not one-to-one.
Only one-to-one functions have inverses. This is because the inverse function must pair each y-value with exactly one x-value, which is only possible if the original function paired each x-value with a unique y-value.
Graphical Representation of Inverse Functions
The graph of an inverse function is a reflection of the original function over the line $y = x$.
This graphical relationship provides a visual way to understand the property $(f \circ f^{-1})(x) = (f^{-1} \circ f)(x) = x$. It shows that composing a function with its inverse (in either order) results in a point on the line $y = x$, which represents the identity function.
ExampleConsider the function $f(x) = 2^x$. Its inverse is $f^{-1}(x) = \log_2(x)$. Their graphs are reflections of each other over the line $y = x$.
Domain and Range of Inverse Functions
The domain of $f^{-1}$ is the range of $f$, and the range of $f^{-1}$ is the domain of $f$. This relationship is a direct consequence of the reflection over $y = x$.
NoteWhen finding an inverse function algebraically, it's crucial to consider the domain and range to ensure the resulting function is valid.
