📉 Functions

🥖 Straight Line

🔁 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 “a 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:

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

👩‍🔬 Modelling