COMPOSITE AND INVERSE FUNCTIONSMrs. Aldous, Mr. Beetz & Mr. ThauvetteIB DP SL Mathematics

You should be able to…

Find the composition of two functions Find the value of the composition of two

functions for a particular value of x Use the graph of a function to decide

whether the function has an inverse Sketch the graph of the inverse of a

function from the graph of a function Verify that two functions are inverses of

each other

You should know…

The composite function can also be denoted by or more simply,

The composition is the operation of applying then ; the composition is the operation of applying then

If is a function that maps to , then the inverse function maps back onto

Geometrically, the graph of the inverse function is a reflection of the original function in the line

You should know…

The domain of is the range of and the range of is the domain of

To find the inverse of a function analytically, simply switch the positions of x and y and then solve the resulting equation for y

A function is the inverse of if and only if

Composite functionsA composite function is made up of two or more functions.

fg(x) means take g(x) and put it into f(x).Replace each x in f(x) with the complete g(x).

gf(x) means take f(x) and put it into g(x).Replace each x in g(x) with the complete f(x).

Try some of these:

Inverse functionsThe inverse of a function, is reversing the operations of that function.

Replace f(x) with ‘y=’

Now make x the subject of the equation.

Finally replace y with x, and the x with a f-1(x).

Another example:

Inverse of quadratic equationsFirst, ensure that the equation is in the completed the square form. Try some of these:

Example

Consider the functions and

(a) Find

(b) Find

(c) Write down the domain of

Example continued…

Example continued…

Example continued…

Be prepared…

The order in which the functions are to be composed is extremely important

Proceed with caution when simplifying expressions resulting from a composition