FunctionsAims: To revise the key skills regarding

functions.

Outcomes

During this session you will. Learn about domain and range and a

functions “personality”. Learn about composite functions. You

will know what fg(x) and how it can be simplified and evaluated.

Finally we will look at inverse functions the properties of inverses and how they are used.

Domain and Range

The Domain is:

The Range is:

Limitations

A functions Domain and Range can be limited in two ways.

1) The Domain may be limited by definition…

If the Domain is limited this may also limit the Range.

2) There may be certain inputs or outputs that are not allowed/will never happen. E.g. x2 never outputs negatives.

50 and ,: 3 xRxxxxf

Personalities

The toughest thing about domain and range is that different functions have different properties that influence domain and range.

For this reason it is good to know a functions “personality”.

Mr Linear

Linear Functions have no functional limitations… you can put any number in and any can come out.

The only way a linear function’s domain or range can be limited is by definition e.g.

53;,23:

.function.. for this range theDefine

xRxxxf

53;,23:

.function.. for this range theDefine

xRxxxf

32;,58:

.function.. for this range theDefine

xRxxxf

Miss Quadratic

The quadratic will accept any x value so has no natural limitations on its domain.

However the range is limited. The range of a quadratic can be

determined by the vertex e.g.

28:

function thisof range thedetermine

point minimum thelocatingfirst By

2 xxxf

28:

function thisof range thedetermine

point minimum thelocatingfirst By

2 xxxf

263:

function thisof range thedetermine

point maximum thelocatingfirst By

xxxf

Mr Root

Root functions may or may not have limited domains and ranges… depends on the root.

You cannot put negative numbers into even roots (nor will negatives come out).

By establishing what values of x will ensure a negative is not being rooted will determine what values can be used in the domain.

Odd roots have no limits on

domain and range

xxf 515:

of...domain thedetermine help torooted

beingfunction for the inequalityan Write

124:

of...domain thedetermine help torooted

beingfunction for the inequalityan Write

xxf

Ms Reciprocal

Functions where there is a division by a variable amount cannot be divided by 0.

Values of x that cause a division by 0 cannot be in the Domain.

Additionally there will likely be an output that requires x=∞ to occur (with 1/x this would be 0)… this output will not be allowed in the range.

Dividing by 0 is a big no no for

any function

52

3:

of...domain thedetermine help to0 equal to

rdenominato thecauses x that of value theFind

xxf

xxf

612

5:

of...domain thedetermine help to0 equal to

rdenominato thecauses x that of value theFind

Mr Sine and Ms Cosine

These wave functions have no numerical limitations on Domain; any value can go in (although we often work within defined x values)

However the sine or cosine element will only ever be between 1 and -1… this often puts limitations on the possible outputs in the range.

1sin3:

of... range thedetermine 1- and 1between values

outputonly cosine and sinefact that thegConsiderin

xxf

2)302cos(4:

of... range thedetermine 1- and 1between values

outputonly cosine and sinefact that thegConsiderin

xxf

Miss Log and Mr Exponential

Logarithms can only be performed on values above 0 (domain x>0) but can output any values (unlimited range).

Exponential functions on the other hand can be performed on any value (unlimited domain) but can only output positive values (f(x)>0)

)13ln(:

of...domain theDetermine

xxf

15:

of... range thedetermine

ations transformgconsiderinBy

3 xexf

Influence of Transformations

That last question highlighted another element that needs to be considered… what transformations do to the domain and range of functions.

For example a translation of -3 in the y direction will have that effect on the range.

Thinking it Through

2:

of... range the writehence 0 is

: of range theknow We

xxf

x

xxf

)2(sin:

of...domain the writehence 11 is

sin: ofdomain theknow We

1

1

xxf

x-

xxf

Dr Modulus

As well as the standard transformations (and combinations of them) you must also be able to incorporate the modulus function.

f(|x|) All negative inputs have the same output as their positive version.

|f(x)| All negative outputs become their positive version.

range. new

thestate and sketch

ofgraph theis Here

xfy

f(x)y

range. new

thestate and )(sketch

ofgraph theis Here

xgy

g(x)y

Composite Functions

gf(x) is an example of a composite function.

gf(x) means perform function g on the output of f.

E.g. If f(x) = x2+2 and g(x) = ln(x) then… gf(x) = ln(x2+2) fg(x) = (lnx)2 + 2 Note that the domain and range can be

influenced by limitations in each function…

Bringing Ideas Together

Extra problems what are the domain and range of gf(x)?

Inverse Functions

An inverse function f-1(x) is a function that for every output in the range of f(x) converts that output back to the input that made it.

E.g.

5: then 5:

2: then 2:

1

1

xxgxxg

xxfxxf

Notable Facts

The graph of an inverse function is a reflection of the original in the line y=x

This means that the domain of the original is the range of the inverse and vice versa.

Making the input of the original the inverse function and the output of the original x you can then rearrange to find the inverse function.

To inverse a function f it must be a 1:1 mapping or have its domain limited so that it is. E.g. sin-1x only inverses sinx for -90≤x≤90

Inverse Example…

)...( find 1

23: 1 xf

x

xxf

To Inverse Quadratic Complete the Square

To inverse a quadratic it must have a limited domain (to right/left of the vertex)

)...( find 3, 26: 12 xfxRxxxxf

)...( find 3, 26: 12 xfxRxxxxf

Working on it…

We will now try numerous examination style problems to reinforce our learning.