 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.

 The Domain is:  The Range is:

 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. x 2 never outputs negatives.

 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”.

 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.

 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.

 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

 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

 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.

 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)

 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.

 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.

 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) = x 2 +2 and g(x) = ln(x) then…  gf(x) = ln(x 2 +2)  fg(x) = (lnx)  Note that the domain and range can be influenced by limitations in each function…

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

 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.

 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 -1 x only inverses sinx for -90≤x≤90

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

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