The Algebra of Functions: Algebra Chapter 2 Section 6

Hello internet, and welcome to the Algebra Lecture Series from the Science of Life.  This time, we are focusing on an introduction to the algebra of functions, or taking two or more functions and applying the algebraic functions in order to combine these functions.
If we have two functions $f(x)$ and $g(x)$ which have the same domain, then applying the same x-value to them and performing a mathematical operation on them yields a single value.  It is important that they either have the same domain or have overlapping domain, since this is the only way that there would be a non-zero value for each function.  For example, a florist cannot physically store a negative quantity of flowers nor an amount of flowers greater than the physical space available.  Therefor, the domain is from 0 to some finite number determined by the physical space.  The same applies for other plants.  So flowers and plants have the same domain.

One mechanism for applying an operation to the functions is to first find the output of each function separately, then applying the operation on those outputs.For example, if $f(x)=x^{2}+3$ and $g(x)=2 \times x+2$ and we wanted to find $f(3)+g(3)$, we first find $f(3)=3^{2}+3=12$ and $g(3)=2 \times 3+2=8$, then add the numbers together, $12+8=20$.  This same concept applies to all basic operations.

There is a certain notation for basic operations of two functions at the same point which you will no doubt come across in your studies of Algebra.  It is having the ([first function][operation][second function])(x).  Here's a table of specifically what I'm getting at:


Operation
Intuitive
Form
Mathematician
Form
Comment
Addition
$f(x)+g(x)$
$(f+g)(x)$

Subtraction
$f(x)-g(x)$
$(f-g)(x)$

Multiplication
$f(x) \times g(x)$
$(f \times g)(x)$

Division
$ \frac {f(x)}{g(x)}$
$\left ( \frac{f}{g} \right )(x)$
$g(x) \neq 0$


Again, these domains must at least overlap in order for the operations to be valid, and the operations are only valid within the overlap.

That's it for Chapter 2.  If you have any questions, please leave them is the comments.  Next chapter will cover how to solve word problems, so subscribe in order to learn this skill.  Hopefully, I can do better than most books and professors at explaining how to do this.  Like and share this post if you found it helpful.  And until next time, stay curious.

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