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πŸ“‰ Functions

πŸ₯– Straight Line

πŸ” Function Notation

Function Introduction:: Take \(5 x^2\). If you start with \(x\) and want to arrive at that result, you must perform a sequene of steps. In this example to get \(5 x^2\) you must 1. square \(x\) making \(x^2\) , then 2. multiply that result by \(5\) making \(5 x^2\). Sequences of steps can sometimes get quite long, so to make things neater instead of writing out all the steps, we give sequences names e.g. \(f(x)\), or \(g(x)\), or \(x(t)\) etc. . In mathematics β€œa function” is a sequence of steps. The notation \(f(x)\) means: take any input number \(x\), do something with that number (e.g. square it then multiply by \(5\)) and this will be the result. For our example equation, \(f(3) = 5 \times 3^2 = 5 \times 9 = 45\). Similarly, \(f(4) = 80\). This lets us name the results of lots of calculations, without having to actually show the details of what sequence of steps we did to do those calculations.

Function Composite:: Nesting functions inside of eachother: \(f(g(x))\) is also written as \(f \circ g (x)\) . First you apply function \(f()\) to \(x\), then you apply function \(g()\) to that result.

Function Inverse:: Going backwards to find what inputs would be needed to arrive at your function’s output. If \(f(x)=2x-3\) then the inverse is \(f'(x) = \frac{y+3}{2}\). Note that any function and its inverse should cancel out to just make \(x\) i.e. \(f^{-1}(f(x)) = x = f(f^{-1}(x))\) .

Function Piecewise:: Gluing together multiple functions sideways, which makes it easy for us to refer to a collection of functions just as a single combo function. For example the absolute value function \(\mid x \mid\) is really this:

\[|x| = \begin{cases} -x & \text{if } x \le 0 \\ \phantom{-}x & \text{if } x \ge 0 \end{cases}\]

πŸ‘©β€πŸ”¬ Modelling

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