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Linear equations

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Quadratic equations

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Functions

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Linear algebra

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Vectors

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Logarithms

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Complex numbers

Useful resources

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Functions

Starter

List out everything you know about functions. This can be things ranging from effects of transformations to things like inverse functions and when/how to get them.

What is a function?

In previous years, you were probably taught that a function is like a set of machines that convert 1 input to 1 output. This still holds true for the IB, however we have to take into account equations that are not functions. By this, we essentially have to see if and when we input 1

x

-value, whether we get a single

y

-value or not. The latter means the equation you've got, is not a function (because it's not possible to have 2

y

-values, otherwise you'd be in some superposition of the two), but is instead just considered a relation. General rules are described in the note below.

Note N03.0a - Tests for functions/relations

Relations in a table

An equation is a function if there's a one-to-one relation with the input-to-output. In other words, no one

x

-value has multiple

y

-values.

$x$	$f(x)$
$0$	$0$
$2$	$1.41$
$2$	$-1.41$
$4$	$2$
$4$	$-2$
Solutions of $x=y^2$

Vertical line test

A test you can do to determine this is the "vertical line test" where you trace a vertical line across the graph, and ensure that vertical line never intersects the graph of the function more than once.

When considering functions, you have to watch out for those that either have a filled in circle (meaning the domain or range is either

\leq

\geq

at the coordinates of the point), or a hollow circle (meaning the domain or range is either

<

>

at the coordinates of the point). If you have a vertical "stack' of filled in circles, you don't have a function (because two values that could be greater than/less than or equal to that

x

-value, mean that you get two

y

-values, and as we know above, that turns out to not be a function).

Domain and range

Every function has a domain and range. The domain represents all the input values of the function (i.e. the "range" of

x

-values you can input, to get an output), whereas the range represents all the values the function can produce as an output.

Topics coming soon:

2) Domain and range

3) Composite functions

4) Inverses

5) Transformations

6) Graphing