1. COMPOSITE AND INVERSE FUNCTIONS Mrs. Aldous, Mr. Beetz & Mr. Thauvette IB DP SL Mathematics

2. 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

3. 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

4. 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

5. Composite functions A 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:

6. Inverse functions The inverse of a function, is reversing the operations of that function. Replace f(x) with ‘ y= ’ Now make x the subject of the equation. -1 Finally replace y with x, and the x with a f -1 (x). Another example:

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

8. Example Consider the functions and (a) Find (b) Find (c) Write down the domain of

9. Example continued…

12. 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