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"
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 \((f \circ g)(x)\).
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 \(\frac{1}{f(x)}\)
\(f(f^{-1}(x)) = x\) - 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 \(y = x\)
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!
Test your understanding with interactive problems covering both composition and inverse functions.