1. Inverse
Functions
Reversing the process

2. Consider the function f(x) = 2x + 1
Set A
Set B –2 5 –3 1 11
The function which changes the numbers in Set B back into the numbers in set A is called an INVERSE function
Make up and example of a relation then define the domain and range of the relation
The opposite of 2x + 1 is take 1 and divide by 2

3. Consider the function f(x) = 2x + 1
Set A
Set B –2 5 –3 1 11
Make up and example of a relation then define the domain and range of the relation
g(x) is the inverse of f(x), usually written as f -1(x)

4. g(x) is the inverse of f(x), usually written as f -1(x)
You have seen this notation before when using your calculator
sin-1, cos-1, tan-1 are the inverse functions of sin, cos and tan
Make up and example of a relation then define the domain and range of the relation

5. Graphically the inverse function is the reflection of the given function in the line y = x [ f(x) = 2x + 1]
Make up and example of a relation then define the domain and range of the relation

6. Exponential & Log
Functions

7. Exponential (to the power of) Graphs
A function in the form
f(x) = ax where a > 0, a ≠ 1
is called an exponential function to base a .
Consider f(x) = 2x x
f(x)
1/ ¼ ½
Remember a0 = 1 and a1 = a

8. Exponential (to the power of) Graphs
y = 2x
A growth function
(1 , 6)
(2 , 4)
y = 6x
(1 , 2)

9. The graph of y = ax always passes through (0 , 1) & (1 , a)

10. y = 2x
y = x
y = log2x

11. To obtain y from x we must ask the question
Log Graphs ie
x 1/8 ¼ ½ y
To obtain y from x we must ask the question
“What power of 2 gives us…?”
This is not practical to write in a formula so we say
“the logarithm to base 2 of x”
y = log2x
or “log to the base 2 of x”

12. y = log2x
(2 , 1)
NB: x > 0
Major Points
(i) y = log2x passes through the points (1,0) & (2,1) .
As x  ∞, y   but at a very slow rate
and as x  0 y  – ∞

13. The graph of y = logax always passes through (1,0) & (a,1)

14. → log 2 8 = 3 → log 10 10000 = 4 → log 6 36 = 2 → log a c = b
You should note carefully the connection between Exponentials and Logs.
Exponential
Log
→ log 2 8 = 3
23 = 8
→ log = 4
104 = 10000
→ log 6 36 = 2
62 = 36
→ log a c = b
ab = c

15. (1 , a)
(a , 1)
(0 , 1)
(1 , 0)

16. Determine the equation of the graph.
The diagram shows the graph of y = f (x) where f is a logarithmic function.
(6 , 1) 1
(4 , 0) +3
Determine the equation of the graph.
y = loga (x – 3)
When x = 6, y = 1
1 = loga 3
 a = 3
y = log3 (x – 3)

17. The diagram shows the graph of y = a log2(x + c)
(6 , 9)
The diagram shows the graph of y = a log2(x + c) –1 1 –2
Determine the values of a and c.
y = a log2 (x + 2)
 c = 3
When x = 6, y = 9
9 = a log2 8
 9 = a × 3
 a = 3

18. The diagram shows part of the graph whose equation is of the form y = 2mx.
What is the value of m?
(3 , 54)
When x = 3, y = 54
54 = 2 × m3
m3 = 27
m = 3

19. Now try
Exercise 7 page 42
1, 2, 3a), 4a) and 8