3. Harder one: A function is defined by : f(x) = (x+3)(x-5) Find the inverse and state the domain and range of f(x) and f -1 (x)

4. A function is defined by : f(x) = (x-16)(x-4) Find the inverse and state the domain and range of f(x) and f -1 (x)

5. Homework: Functions on moodle with mark scheme

6. Inverse functions  Recap of inverse of a function.  Inverse functions with e x and ln x  Harder inverse functions such as quadratics and algebraic fractions

7. A neat little trick… ► As always in maths, there is a trick to this… 1. Write function as a rule in terms of y and x. 2. Swap ‘x’ and ‘y’ 3. Rearrange to get in terms of y. 4. Write as f -1 (x) =

8. A neat little trick… Find the inverse of

9. Inverse functions Inverse functions only exist for one-one functions.

10. Things to note..  The domain of f -1 is the range of f and the range of f -1 is the domain of f.  The graph of an inverse function can be found by reflecting a function in the line y=x

11. Have a go: Worksheet C Questions 1-7 ppt: 10 questions

12. y=ln(x) is a reflection of y = e x in the line y = x y = e x y = x y = ln (x) y = e x, y = x and y = ln x y=ln(x) and y = e x are inverse functions

13. Inverse functions with e x e.g. f(x) = e x -2 x = e y -2 x +2 = e y ln(x +2) = ln e y ln(x +2) = y The inverse of f(x) is … f -1 (x) = ln(x +2) Domain is x > -2 y = e x - 2 y = ln (x+2) “The graph of an inverse function can be found by reflecting a function in the line y=x”

14. Inverse functions with e x e.g. f(x) = e 2x-1 + 6 x = e 2y-1 + 6 x - 6 = e 2y-1 ln(x - 6) = ln e 2y-1 ln(x - 6) = 2y - 1 The inverse of f(x) is … f -1 (x) = ½(ln(x-6) + 1) Domain ? ln(x - 6) +1 = 2y ½(ln(x - 6) +1) = y Domain is x > 6 Cannot have ln of numbers less than 0

15. Inverse functions with ln x e.g. f(x) = ln(2x) + 6 x = ln(2y) + 6 x - 6 = ln (2y) e x-6 = e ln 2y e x-6 = 2y The inverse of f(x) is … f -1 (x) = ½ e x-6 Domain ? ½ e x-6 = y

16. Have a Go Domino Trail or Worksheet C Questions 8,9,11 Extension: Exam Questions ppt: 10 questions

17. Plenary

19. Plenary

29. f(x) = x. x+1 Show that f -1 (x) = 1 - 1 1-x Extension: