Functions in Math

[Property Is Robbery]

I'm taking a trig class at community college and I was wondering if someone could explain to me, in layman's terms, what is a function and how to solve one. Thanks :)

[tbasherizer]

A function is any mathematical operation being carried out on a value you plug in.

For example, if I've got a function f(x) = x + 2, the function just adds to to whatever x is. so in this case, f(2) = 4.

In reverse, if you're told that f(x) = x + 2 and that f(3) = 5, you just make f(x) y and solve it using algebra. So x in this case would be 3.

If you're doing trigonometry, I'm sure it's more complicated than the above. There's a more mathy definition than what I gave, but that should do you well for at least a cursory understanding. Tell me if this works for you purposes.

[Property Is Robbery]

Thats a good definition for now. How do domain and range fit into that?

(I tested well in my placement exam for college but I had some shitty teachers in high school so I feel behind)

[Broletariat]

Domain is all the X values that can "work" within a function.

If you have the function.

f(x) = 1/x

Then the number 0 does not work as an X value, therefor the domain is all numbers not 0.

The range is every number that the Y value could be. If your function was f(x) = |x|

Then your range would be all positive numbers and 0, no negatives at all.

If you want any help feel free to PM me, I'm planning on being a a math teacher in high school and am currently taking Calc 2.

[thefinalmarch]

How on earth do math classes work in the US? Here in Australia I'm taking a so-called "Mathematical Methods" class which includes calculus, trigonometry, and probability. Yet, in the US there seems to be a separate class for each broad category of math.

[Broletariat]

How on earth do math classes work in the US? Here in Australia I'm taking a so-called "Mathematical Methods" class which includes calculus, trigonometry, and probability. Yet, in the US there seems to be a separate class for each broad category of math.

Up until secondary education our math is mostly rudimentary arithmetic and maybe a slight smidge of algebra.

In secondary education the classes usually go like this.

Algebra 1 > Algebra 2 > Geometry > Pre-calculus > Calculus.

Pre-calculus introduces a lot of the trig that Calculus will work with but it is also examined in Geometry. Geometry and Algebra 2 are usually interchangeable.

Algebra 1 deals with very basic algebra, solving for x, learning to factor and FOIL etc.

Algebra 2 deals with more advanced factoring methods like completing the square, the quadratic formula etc.

Calculus 1 is an introduction to limits, derivatives and integrals. You learn the power rule forwards and backwards, quotient/product rule along with the chain rule. My high school taught us the backwards chain rule as well, but my calc 2 professor seems to think it is calc 2 material, along with left/right/middle rectangles and the trapezoidal method, all of which I covered in my AP Calculus class.


There's also a whole separate class on Statistics which is typically one of the lesser taken math classes in high school.

[thefinalmarch]

What I'm doing is probably equivalent to calc 1 then. Done limits and derivatives, going to do integrals whatever the fuck they are in a month or two (at the moment we're doing trig, although we covered the basics last year). Also I guess you could have counted the first half of the year as being algebra 2 seeing as we only learnt completing the square and the quadratic formula then (also polynomial division and a horde of other shit). I get the feeling math classes here are far more condensed time-wise than in 'murika.

[thefinalmarch]

Also we have a "specialist math" class. I'm pretty sure it's also almost entirely statistics though.

[CommunityBeliever]

what is a function

A function is a set of order pairs { (input, output) } such that [ #(arg, ?) = 1 ], in other words for each input there is exactly one output value. For notational elegance we can express this as { input →output }.

One of the simplest functions is not (also known as the logical negation) which takes in one truth value and returns the opposite one:



false → true
true → falseAn slightly more complicated function is or (also known as logical disjunction) which outputs true if either one of its arguments are true:


(false, false) → false
(false, true ) → true
(true, false) → true
(true, true ) → trueA similar function is and (also known as logical conjunction):


(false, false) → false
(false, true ) → false
(true, false) → false
(true, true ) → trueThese are examples of simple functions that are defined by listing out all possible values. In practice our physical limitations prevent us from listing everything out, so we define things intensionally by using trees of symbols:

http://www.redhat.com/magazine/002dec04/features/gcc/figs/ast.png

Even more complication comes form mathematical notation, which uses infix, postfix, prefix, and many other syntactic forms. As such we also need sophisticated parser programs (https://github.com/rakudo/rakudo/blob/master/src/Perl6/Grammar.pm) to turn it back into a syntax tree like above:


x = 3 * (y - z)After all this we can finally write out mathematical notation to define more complicated functions.


Thats a good definition for now. How do domain and range fit into that?The domain is the set of all inputs and the range is the set of all outputs. In sense this is a property of every function F:

F ⊆ (domain × range)


how to solve oneIt depends, what are you solving for?