03

Functions

\Delta _{\beta \alpha}

What is a function?03_01A

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.

Tests for functions and 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 range03_02A

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).

Domain and range rules for 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.

Finding ranges and domains

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 functions03_03A

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

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.

Transformations03_04A

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.

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

Reflection over the

x

-axis: in the form

-f(x)

Reflection over the

y

-axis: in the form

f(-x)

Vertical stretch for

|a|>0

: in the form

af(x)

Vertical compression for

0<|a|<1

: in the form

af(x)

Horizontal compression for

|b|>0

: in the form

f(bx)

Horizontal stretch for

0<|b|<1

: in the form

f(bx)

New function translations

Rational transformations: in the form

\frac{1}{f(x)}

Trigonometric transformations: in the form

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

Rational transformations03_05A

Rational transformations essentially act like stretches/compressions in the

y

-axis. For any given function, you put

1

over the output (seen in the notation above). However when you get to graphing them in a few sections, you'll have to be careful about asymptotes. Any time you divide by

0

, you get an undefined output. Therefore, whatever

x

-value you input that causes the output (or denominator) to be

0

is where a vertical asymptote will be at. This is because the function can get closer and closer either from above or below

0

, but as soon as it reaches

0

, it's unable to exist anymore!

Trigonometric translations/transformations03_06A

Trigonometric function transformations have similar formats to those of regular functions, such as up/down and left/right transformations, but there's also period and phase transformations described and shown below.

Trig translation and transformation formats

For

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

a

- affects the amplitude of the function (the "height" relative to the principal axis)

b

- affects the period of the function (the period is defined as

\frac{2\pi}{b}

)

c

- affects the horizontal translation of the function (same as regular transformations)

d

- affects the vertical translation of the function (same as regular transformations)

Topics coming soon

Composite functions coursework
Inverses
Transformations
Graphing functions
Practical applications
ECQs

Linear functions

Quadratic functions

Functions

Trigonometry

Trigonometric functions

Linear algebra

Vectors

Logarithms

Complex numbers

Differentiation

Integration

03

What is a function?03_01A

Tests for functions and relations

Domain and range03_02A

Domain and range rules for functions

Finding ranges and domains

Composite functions03_03A

Finding function composites

Transformations03_04A

Function translations

Rational transformations03_05A

Trigonometric translations/transformations03_06A

Trig translation and transformation formats

Topics coming soon