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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

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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 (as in, test out

x=1

x=2

, etc...).

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. These attributes describe the "dimensions" per-se of a function. The domain represents all the input values of the function (the

x

-values), whereas the range represents all the values the function can produce as an output (the

y

-values). When looking at the domain, you have to find all the solutions which make the function "work", as in, you can't divide by

0

in a rational, and you can"s square root a surd (you"ll learn why in the complex numbers chapter).

Note N03.1a - Rules for domains/ranges of functions

Domain

1) In rational functions: domains cannot equal 0. Example:

f(x)=\frac{1}{x-2}

x\neq 2

2) In surd functions: inner expression must be non-negative. Example:

g(x)=\sqrt{x+3}

x\geq-3

3) In logarithmic functions: inner expression must be positive. Example:

h(x)=\log(x-1)

x>1

4) There are typically no restrictions on the domain of a polynomial function (linear, quadratic, cubic, etc...)

Range

1) Substitute values from the domain into the function

2) Analyze behavior at extreme values (limits approaching infinity)

3) Use graphical methods or algebraic manipulation to find maximum and minimum values

Further in this chapter, you will see that we use domain and range restrictions so that we can create inverses of functions. You will also learn how to graph those inverses, even if you"re dealing with a relation.

Coursework C03.0a - Find the functions and range/domain

1) Find the domains and ranges of the functions below.

f(x)=\sqrt{2x-6}

g(x)=\frac{1}{x^2-4}

h(x)=x^2-2x+1

2) For the sets of points below, determine if they're functions or relations.

A={(1,2),(1,3),(2,4)}

B={(0,1),(0,2),(1,3)}

C={(a,b),(a,c),(d,e)}

3) Determine whether these sets of points are a function or not, and if they are, what the domain and range are.

D={(2,3),(3,4),(4,5)}

E={(0,4),(2,8),(-2,8)}

F={(2,3),(3,4),(2,5)}

Composite functions

Composite functions are ones that use multiple functions together at once, to essentially calculate the output all in one go, instead of calculating it through multiple steps. The general notation is as follows:

h(x)=(g\circ f)(x)

, where you take function

f(x)

and put it into function

g(x)

to produce the output. Note, you can also express it as

g(f(x))

if you need to find a function, given one and the composite.

When considering domains, for

g\circ f

, the output of

f(x)

must be within the domain of

g

Example E03.0a - Finding function composites

Given two functions

1) Let

f(x)=2x+3

and

g(x)=x^2-1

. Find

(g\circ f)(x)

Just sub in

f(x)

into

g(x)

like so:

(2x+3)^2-1

That simplified gets you

4x^2

Given one function, and composite

2) Given the function

3x^2+18x+20

, first find it in the form

a(x+b)^2+c

, then given that

f(x)=x+3

and

(g \circ f)(x)=3x^2+18x+20

, find

g(x)

This question nicely guides you through what you have to do! First find the vertex form, from which we can figure out what's "applied" to it to get to that composite function

In vertex form, we get

3(x+3)^2-7

. From here, we can look at what happens to

f(x)=x+3

to get to that vertex form. It's just multiplying by

3

, squaring that, and subtracting

7

\therefore g(x)=3x^2-7

Composite functions should be self explanatory, so there wont be much discussion past the above.

Transformations

You can translate/scale many different functions in many different ways, but there are some key "formats" per-se, that you should learn, in order to transform a function or to be able to graph one if you're given one with numerous translations.

In the IB, you only really have to worry about adding these two new "formats": rational and trigonometric translations. The rest, you should know (such as up-down, left-right, and scaling), though I will still add them below.

Note N03.0a - Function translations

Preexisting knowledge

Up-down translations: in the form

f(x)+k

and

f(x)-k

respectively

Left-right translations: in the form

f(x+h)

and

f(x-h)

respectively

Scale in the

x

-axis: in the form

Scale in the

y

-axis: in the form

New function translations

Rational transformations: in the form

\frac{1}{f(x)}

Trigonometric transformations: in the form

a\sin{(b(x-c))}+d

Rational transformations

Rational transformations essentially act as stretching ones, whereby you take the

y

output from a function and put 1 over that. If you think about it, what happens when you divide by

0

? It is undefined, right? Will in that case, at the

x

-coordinate that created that output, we have a vertical asymptote (the function never reaches that "line" because it just does not exist there).

Trigonometric transformations

Trigonometric functions also have a "layout" that is universal (between

\sin

and

\cos

) in determining how it should be translated. The

a

value affects the amplitude of the function (the "height" from the principal axis to either a peak or trough). The

b

value affects the period of the function (the period is

\frac{2\pi}{b}

and determines how many cycles of the function are in one second). The

c

value affects horizontal translation, and lastly the

d

value affects vertical translation.

Topics coming soon:

3) Composite functions coursework

4) Inverses

5) Transformations

6) Graphing