# More on Functions

Welcome to the third week’s session for MATH 107! This week, we are looking at functions in greater detail and exploring some of their properties. You will look at minima and maxima, transformations, and relations of multiple functions. So let’s get started!

## Domain and Range

The domain and range are two of the simplest properties of a function. The domain represents all possible values for $x$. This is the permissible inputs to a function, but that is usually any number. However, there can be exceptions to that.

The range is more complicated. It is constrained by the operation within the function. For a purely linear function, that would be any number again. But it gets complicated quickly. Watch this video for more information.

Check this out to interactive to test the domain and range of a function. For instance, try $y = \sin{x}$.

## Operations on Functions

Operations, like addition or multiplication, can be done on functions just like on numbers. This gives us a set of tools we can use to understand how functions relate to each other. The most complicated of these are the composition of functions. Watch this video for more information.

In addition, this interactive demonstration shows how composition
works for two simple functions.

[WolframCDF source=”http://demonstrations.wolfram.com/CompositionOfFunctions/CompositionOfFunctions.cdf” width=”600″ height=”650″]

Construct $f(g(x))$ by starting at a point on the $x$ axis and moving vertically to the graph of $g(x)$. Move horizontally to the graph of $y=x$ to relocate $g(x)$ to an $x$ value. Next evaluate $f$ there by moving vertically to the graph of $f(x)$. Finally move horizontally to the point $(x, f(g(x)))$.

## Transformations

Function transformations allow us to move functions around on the graph. But transformations are just a special case of other function operations. Making this connection now makes this easier. Watch this video:

Now, you can test a function transformation here:

[WolframCDF source=”http://demonstrations.wolfram.com/FunctionTransformations/FunctionTransformations.cdf” width=”600″ height=”615″]