> For the complete documentation index, see [llms.txt](https://jamesbrayy.gitbook.io/atar/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://jamesbrayy.gitbook.io/atar/specialist/functions.md).

# functions

***

## **composition**

* the domain of $$f\circ g\left(x\right)$$ is all $$x$$ where $$g\left(x\right)$$ is in the domain of $$f$$ and $$x$$ is in the domain of $$g$$
  * *i.e.* $$dom\left(f\circ g\right)=\left{x\in dom\left(g\right)\ :\ g\left(x\right)\in dom\left(f\right)\right}$$
* the range of $$f\circ g\left(x\right)$$ is all values that $$f\circ g\left(x\right)$$ takes as $$x$$ runs over its domain
  * *i.e.* $$ran\left(f\circ g\right)=\left{f\circ g\left(x\right)\ :x\in dom\left(f\circ g\right)\right}$$

## **inverse**

* inverse functions are geometric reflections across the line $$y=x$$

#### *domain and range:*

* the domain of $${f}^{-1}\left(x\right)$$ is the same as the range of $$f\left(x\right)$$, but $$f(x)$$ must be restricted a one-to-one (injective) portion of its domain to ensure $${f}^{-1}$$ is a function
  * *i.e.* $$dom\left({f}^{-1}\right)=ran\left(f\right), \quad f \text{ restricted to be injective}$$
* the range of $${f}^{-1}\left(x\right)$$ is the same as the domain of $$f\left(x\right)$$ (with restrictions if required)
  * *i.e.* $$ran\left({f}^{-1}\right)=dom\left(f\right), \quad f \text{ restricted to be injective}$$

## **reciprocal**

#### *domain and range:*

* the domain of $$\frac{1}{f\left(x\right)}$$ is all $$x$$ where $$f\left(x\right)$$ is defined and not equal to $$0$$
  * *i.e.* $$dom\left(\frac{1}{f}\right)=\left{x\in dom\left(f\right)\ :f\left(x\right)\ne 0\right}$$
* the range of $$\frac{1}{f\left(x\right)}$$ is all values that $$\frac{1}{f\left(x\right)}$$ takes as $$x$$ runs across its domain
  * *i.e.* $$ran\left(\frac{1}{f}\right)=\left{\frac{1}{f\left(x\right)}\ :x\in dom\left(f\right),\ f\left(x\right)\ne 0\right}$$

## **domain**

1. $$y = \frac{1}{ax + b}$$
   * set $$ax+b=0$$ and solve for x
2. $$y = \sqrt{ax + b}$$
   * set $$ax+b\ge 0$$ and solve for x
3. $$y = \sqrt{ax^2 + bx + c}$$
   * set $$a{x}^{2}+bx+c\ge 0$$, factor and solve for x
4. $$y = \log\_a(bx + c)$$
   * set $$bx+c>0$$ and solve for x
5. $$y = \frac{1}{\sqrt{ax + b}}$$
   * set $$ax+b>0$$ and solve for x
6. $$y = \frac{1}{\sqrt{ax^2 + bx + c}}$$
   * set $$a{x}^{2}+bx+c>0$$, factor and solve for x