1. f(x) = x2 x > 3
Find the range of f(x)
f(x) > 9

2. Find f-1(x) and state its domain
f(x) = x2 + 6x x > 0
Find f-1(x) and state its domain
𝑓 −1 𝑥 = 𝑥+1 −3
Domain x > 8

3. f(x) = x2 – g(x) = e2x
Find
gf(x)
fg(x)
𝑒 2 𝑥 2 −6
𝑒 4𝑥 −3

4. Define the transformations
f(x) = x2
Define the transformations
𝑓 𝑥 =2− 𝑥 2
𝑡𝑟𝑎𝑛𝑠𝑙𝑎𝑡𝑖𝑜𝑛 0 −2 𝑓𝑜𝑙𝑙𝑜𝑤𝑒𝑑 𝑏𝑦 𝑎 𝑟𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛 𝑖𝑛 𝑡ℎ𝑒 𝑥−𝑎𝑥𝑖𝑠 or
𝑟𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛 𝑖𝑛 𝑡ℎ𝑒 𝑥−𝑎𝑥𝑖𝑠 𝑓𝑜𝑙𝑙𝑜𝑤𝑒𝑑 𝑏𝑦 𝑎 𝑡𝑟𝑎𝑛𝑠𝑙𝑎𝑡𝑖𝑜𝑛 0 2

5. Define the transformations
f(x) = x2
Define the transformations
𝑓 𝑥 =2 𝑥 2 −4
𝑡𝑟𝑎𝑛𝑠𝑙𝑎𝑡𝑖𝑜𝑛 0 −2 𝑓𝑜𝑙𝑙𝑜𝑤𝑒𝑑 𝑏𝑦 𝑎 𝑠𝑡𝑟𝑒𝑡𝑐ℎ 𝑓𝑎𝑐𝑡𝑜𝑟 2 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙 𝑡𝑜 𝑡ℎ𝑒 𝑦−𝑎𝑥𝑖𝑠 Or
𝑠𝑡𝑟𝑒𝑡𝑐ℎ 𝑠𝑐𝑎𝑙𝑒 𝑓𝑎𝑐𝑡𝑜𝑟 2 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙 𝑡𝑜 𝑡ℎ𝑒 𝑦−𝑎𝑥𝑖𝑠 𝑓𝑜𝑙𝑙𝑜𝑤𝑒𝑑 𝑏𝑦 𝑎 𝑡𝑟𝑎𝑛𝑠𝑙𝑎𝑡𝑖𝑜𝑛 0 −4

6. Differentiate with respect to x
𝑒 𝑥 2
2cos ½x
e3x
−sin⁡ 𝑥 2
3 𝑒 3𝑥 ⁡
2𝑥 𝑒 𝑥 2 ⁡

7. Sketch the graph of
y = 2 𝑒 −2𝑥

8. Sketch the graph of
y = 3ln(x+2)

9. Sketch the graph of
y = |ln x−2 |

10. Sketch the graph of
y = |9 – 4x2|

11. Sketch the graph of
y = e|2x|

12. Differentiate with respect to x
− 2𝑥 2− 𝑥 2 ⁡
2 𝑥 ⁡
18x(3x2+4)2⁡
(3x2 + 4)3
ln (2- x2)
ln (3x2)
e5-x
− 𝑒 5−𝑥

13. x2e2x = 2xe2x Solve 𝑥 2 𝑒 2𝑥 𝑥 𝑒 2𝑥 =2 so x = 2
There are two solutions so what is wrong with this method?
𝑥 2 𝑒 2𝑥 𝑥 𝑒 2𝑥 = so x = 2

14. Solve
x2e2x = 2xe2x
x2e2x - 2xe2x = 0
xe2x (x – 2)=0
x = 0 x = 2

15. Sketch the graph of
y = sin-1(x)

16. Sketch the graph of
y = cos-1(x-1)

17. y = 𝑒 3𝑥 ( 𝑥 2 −2) 1 3
Find 𝑑𝑦 𝑑𝑥
2 3 𝑥𝑒 3𝑥 ( 𝑥 2 −2) − 𝑒 3𝑥 ( 𝑥 2 −2) 1 3

18. y = 6𝑥−1 𝑠𝑖𝑛2𝑥
Find 𝑑𝑦 𝑑𝑥
6𝑠𝑖𝑛2𝑥−2 6𝑥−1 𝑐𝑜𝑠2𝑥 𝑠𝑖𝑛 2 2𝑥

19. 𝐷𝑜𝑛′𝑡 𝑓𝑜𝑟𝑔𝑒𝑡 +𝑐 2 2𝑥+3 4sin 2x e3x 1 3 𝑒 3𝑥 −2𝑐𝑜𝑠2𝑥 ln⁡|2𝑥+3|
Integrate with respect to x
2 2𝑥+3
4sin 2x
e3x
1 3 𝑒 3𝑥
−2𝑐𝑜𝑠2𝑥
ln⁡|2𝑥+3|
𝐷𝑜𝑛′𝑡 𝑓𝑜𝑟𝑔𝑒𝑡 +𝑐

20. 𝐷𝑜𝑛′𝑡 𝑓𝑜𝑟𝑔𝑒𝑡 +𝑐 4𝑥 (1+𝑥 2 ) 1 1−2𝑥 4 (1+2𝑥) 2 − 2 1+2𝑥 2ln⁡(1+ 𝑥 2 )
Integrate with respect to x
− 2 1+2𝑥
2ln⁡(1+ 𝑥 2 )
− 1 2 ln⁡|1−2𝑥|
4𝑥 (1+𝑥 2 )
1 1−2𝑥
4 (1+2𝑥) 2
𝐷𝑜𝑛′𝑡 𝑓𝑜𝑟𝑔𝑒𝑡 +𝑐

21. Don’t forget the final statement
y = 3ln(3e – x) intersects the line y = x at x = ∝ . Show that ∝ between 4 and 5
Don’t forget the final statement

22. STAIRCASE x1 x2 x3

23. Numerical Integration Simpsons rule 0 𝜋 𝑥𝑠𝑖𝑛2𝑥 𝑑𝑥 with 5 ordinates
Is your calculator in radians?
If possible work with exact values….
Show values in a table.
Write the formula out with your values substituted in

24. Numerical Integration Mid-ordinate rule 0 𝜋 𝑥𝑠𝑖𝑛2𝑥 𝑑𝑥 with 4 strips
What are the x-values you would use ?