Composite & Inverse Functions

Function Composition

Function composition is like a chain of operations where one function's output becomes another function's input. Think of it as a mathematical assembly line!

For functions \(f(x)\) and \(g(x)\), their composition is written as:

\((f \circ g)(x) = f(g(x))\)

Read as "f composed with g of x" or "f of g of x"

Example

Let \(f(x) = x^2\) and \(g(x) = x + 1\)

Then \((f \circ g)(x) = f(g(x)) = f(x + 1) = (x + 1)^2\)

So if we input 2:

    First, \(g(2) = 2 + 1 = 3\)

    Then, \(f(3) = 3^2 = 9\)

    Therefore, \((f \circ g)(2) = 9\)

Interactive Graph: Watch how composition works! The blue curve shows \(f(x) = x^2\), the red curve shows \(g(x) = x + 1\), and the green curve shows their composition \((f \circ g)(x)\).

Inverse Functions

An inverse function reverses what the original function does. If a function is like following a map forward, its inverse is like following the map backward!

For a function \(f(x)\), its inverse \(f^{-1}(x)\) "undoes" the original function:

    If \(f(a) = b\), then \(f^{-1}(b) = a\)

    The \(^{-1}\) notation doesn't mean \(\frac{1}{f(x)}\)

Key Properties

    \(f(f^{-1}(x)) = x\) - Going forward then backward gets you back to start

    \(f^{-1}(f(x)) = x\) - Going backward then forward gets you back to start

    The graph of \(f^{-1}(x)\) is a reflection of \(f(x)\) over the line \(y = x\)

Example

Let \(f(x) = 2^x\)

Its inverse is \(f^{-1}(x) = \log_2(x)\)

Check: If \(f(3) = 2^3 = 8\), then \(f^{-1}(8) = \log_2(8) = 3\)

Interactive Graph: The blue curve shows \(f(x) = 2^x\), the red curve shows its inverse \(f^{-1}(x) = \log_2(x)\), and the dashed line is \(y = x\). Notice how they reflect across this line!

Ready to Practice?

Test your understanding with interactive problems covering both composition and inverse functions.