(f circ g)(x) = f(g(x)): plug g(x) into f. Important: f(g(x)) neq g(f(x)) in general! Order matters. Example: f(x) = x^2, g(x) = x + 3. - f(g(x)) = f(x+3)…
(f circ g)(x) = f(g(x)): plug g(x) into f. Important: f(g(x)) neq g(f(x)) in general! Order matters. Example: f(x) = x^2, g(x) = x + 3. - f(g(x)) = f(x+3) = (x+3)^2 - g(f(x)) = g(x^2) = x^2 + 3 Different results!
Example: If f(x) = 2x + 1 and g(x) = x - 4, find f(g(3)).
The inverse f^{-1}(x) undoes f: f(f^{-1}(x)) = x. To find f^{-1}: 1. Write y = f(x) 2. Swap x and y 3. Solve for y Key property: The graph of f^{-1} is the reflection of f across y = x. If (a, b) is on f, then (b, a) is on f^{-1}.
Example: Find the inverse of f(x) = 3x - 6.
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