This subject covers the essential concepts and applications of functions in high school mathematics, preparing students for advanced mathematical reasoning.
Two advanced ideas are inverse functions and composite functions.
An inverse function undoes the action of the original. If \( f(x) \) turns \( x \) into \( y \), then the inverse \( f^{-1}(x) \) turns \( y \) back into \( x \).
You can combine two functions into one by plugging the output of one into the input of another. This is called composition, written as \( (f \circ g)(x) = f(g(x)) \).
These ideas are super useful in math and science, especially when solving equations or modeling real-life processes that happen in steps.
If \( f(x) = 2x + 3 \), its inverse is \( f^{-1}(x) = \frac{x-3}{2} \).
If \( f(x) = x^2 \) and \( g(x) = x + 1 \), then \( f(g(x)) = (x+1)^2 \).
Inverses undo functions; composites mix multiple functions together.