1. Higher
Functions
Unit 1.2

2. A rule which changes one number into another.
What is a FUNCTION
A rule which changes one number into another.
A rule which links each member of Set A to EXACTLY ONE member of Set B.

3. Domain and Range
The values that make up the set of independent values are the domain
The values that make up the set of dependent values are the range.
Make up and example of a relation then define the domain and range of the relation

4. Consider the function f(x) = 2x + 1
The values that make up the set of independent values are the domain
Let x = -2 , 0 , 5 (the domain)
The values that make up the set of dependent values are the range.
Make up and example of a relation then define the domain and range of the relation
f(x) = -3 , 1 , 11 (the range)

5. 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
Domain
Range

6. Some Things About Functions
1. Remember that for each member of the DOMAIN there is EXACTLY one value in the RANGE.
2. Sometimes you have to restrict the numbers in the DOMAIN depending on the function you are working with.

7. Pick some values for x, the independent values
Domain and Range
To draw a graph of y = 3x + 1
Pick some values for x, the independent values
Use the values for x in y = 3x + 1 to
find the corresponding values of y, the dependent values.
Make up and example of a relation then define the domain and range of the relation
The y values depend on the values chosen for x which are independent

8. Functions - Notation
There are two ways to write a function but they both have the same effect
The function on the left is read as follows:-
‘Function f such that x is mapped onto x2 + 4’
The function on the right is read as follows:-
‘f of x equals x2 + 4’

9. Suppose that f(x) = 3x - 2 and g(x) = x2 +1
Example 1
f(2) = 3×2 – 2 = 4
Suppose that f(x) = 3x and g(x) = x2 +1
g(4) = =17
g(2) = = 5
f(5) = 5×3-2 = 13
f(1) = 3×1 - 2 =1
f(1) = 3×1 - 2 =1
(a) g( f(2) ) =
g(4)
= 17
g(26) = = 677
g(5) = = 26
(b) f( g (2) ) =
f(5)
= 13
(c) f( f(1) ) =
f(1)
= 1
(d) g( g(5) )
= g(26)
= 677

10. Suppose that f(x) = x + 5 and g(x) = x2 – 3
Example 1
Suppose that f(x) = x and g(x) = x2 – 3
(a) g( f(1) ) =
g(6)
= 33
(b) f( g (1) ) =
f(– 2)
= 3
(c) f( f(3) ) =
f(8)
= 13
(d) g( g(2) )
= g(1)
= – 2

11. Suppose that f(x) = x + 3 and g(x) = 2x – 3
Example 2 finding a formula
Suppose that f(x) = x and g(x) = 2x – 3
(a) g( f(x) ) =
g(x + 3 )
= 2(x + 3) – 3
= 2x + 6 – 3
= 2x + 3
(b) f( g(x) ) =
f(2x – 3)
= (2x – 3) + 3
= 2x – 3 + 3
= 2x

12. We need to revise some algebra before going any further
There is a quicker way to square brackets than using FOIL
(x + a)2 = (x + a)(x + a)
= x2
+ ax
+ ax
+ a2
1. Square each term
= x2 + 2ax + a2
x a2
2. Multiply the two terms then double your answer
x2 + 2ax + a2

13. 2. Multiply the two terms then double your answer
(x + a)2
(x + 5)2
1. Square each term
x a2 x
2. Multiply the two terms then double your answer
x2 + 2ax + a2
x x
2 × 5 × x

14. 2. Multiply the two terms then double your answer
(3x – 2)2
1. Square each term
x a2 9x
2. Multiply the two terms then double your answer
x2 + 2ax + a2
9x2 – 12x + 4
2 × (–2) × 3x

15. Suppose that f(x) = 3x - 2 and g(x) = x2 +1
Find formulae for (a) g(f(x)) (b) f(g(x))
(a) g(f(x)) = g(3x – 2)
= (3x – 2)2 + 1
= 9x2 – 12x + 5
(b) f(g(x)) = f(x2 + 1)
= 3(x2 + 1) – 2
= 3x2 + 1
NB: g(f(x))  f(g(x)) in general.
CHECK
g(f(2)) =
9 × × 2 + 5 =
= 17
f(g(2)) =
3 ×
= 13

16. Page 24
4, 5, 6, 7 up to 7d

17. Let h(x) = x – 3, g(x) = x2 + 4 and k(x) = g(h(x))
If k(x) = 8 then find the value(s) of x
k(x) = g(h(x))
Put x2 – 6x = 8
= g(x – 3)
then x2 – 6x + 5 = 0
= (x – 3)2 + 4
or (x – 5)(x – 1) = 0
= x2 – 6x + 13
So x = 1 or x = 5
CHECK
g(h(5))
= g(2) =
= 8

18. Choosing a suitable Domain
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
Domain
Range

19. Choosing a Suitable Domain
There are times when a function f(x) does not exist for specific values of x.
In particular, when the function involves algebraic fractions or square roots.
A fraction does not exist when the denominator is zero
A square root does not exist for values less than zero

20. Choosing a Suitable Domain
(i) Suppose f(x) = 1
x2 – 4
f(x) is not defined when x2 – 4 is zero
x2 – 4 = 0
 is the set of Real Numbers
So x2 = 4
So x = –2 or 2
Hence domain is {x   : x  –2 or 2 }
f(x) exists for all values of x except when x = –2 or 2

21. Choosing a Suitable Domain
Suppose f(x) = 1
x2 – 1
and g(x) = 2x
function is not defined when (2x)2 – 1 is zero
 (2x + 1)(2x – 1) = 0
 x = ± ½
f(g(x)) exists for all x except x = ± ½

22. f(x) exists only if 2x – 1 is greater than or equal to zero
Choosing a Suitable Domain
f(x) exists only if 2x – 1 is greater than or equal to zero
2x – 1 ≥ 0
2x ≥ 1
x ≥ 1/2
f(x) exists only if x ≥ 1/2

23. Functions with Fractions
To simplify expressions like this change all parts to the same denominator
x ≠ 1

24. f(f(x)) exists for all x except x = – 2
Suppose f(x) = 1
x + 1
Find an expression for f(f(x)) and a suitable domain.
f(f(x)) exists for all x except x = – 2

25. Always give suitable Domain
Page 20 ~ 8 to 12
Page 24 ~ 7f) to 9
Always give suitable Domain

26. Many examples require you to sketch a parabola to find where values are positive or negative

27. Greater than or equal to zero when x ≤ -4 and x ≥ 2
(ii) Suppose that g(x) = (x2 + 2x - 8)
We need (x2 + 2x – 8)  0 -4 2
Suppose x2 + 2x – 8 = 0
Then (x + 4)(x – 2) = 0
So x = – 4 or x = 2
A “happy” parabola
So domain = { x   : x  – 4 or x  2 }

28. x ≠ 0
x ≠ – 3

29. Find f(g(x)) and g(f(x)) in their simplest form when
Exercise
Find f(g(x)) and g(f(x)) in their simplest form when
Find f(g(x)) and g(f(x)) in their simplest form when

33. Inverse
Functions
Reversing the process

34. 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

35. 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)

36. 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

37. 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

38. Exponential & Log
Functions

39. 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

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

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

42. 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”

43. 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  – ∞

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

45. The following questions are on
Graphs & Functons
Non-calculator questions will be indicated

46. Functions and are defined on
suitable domains.
a) Find an expression for h(x) where h(x) = f(g(x)).
b) Write down any restrictions on the domain of h. a) b)

47. a) Find expressions for: i) f(g(x)) ii) g(f(x))
Functions f and g are defined on suitable domains by f(x)= sin x and g(x) = 2x.
a) Find expressions for:
i) f(g(x)) ii) g(f(x))
b) Solve 2 f(g(x)) = g(f(x)) for 0 ≤ x ≤ 360 a) b)

48. Functions are defined on a suitable set of real numbers.
a) Find expressions for
b) i) Show that
ii) Find a similar expression for
and hence solve the equation a) b)
Now use exact values
equation reduces to

49. a) b) c) from the graph, k  4
A sketch of the graph of y = f(x).
The graph has a maximum at A and a minimum at B(3, 0)
a) Find the co-ordinates of the turning point at A. b) Hence, sketch the graph of
Indicate the co-ordinates of the turning points. c) Write down the range of values of k for which
g(x) = k has 3 real roots. a)
Differentiate
for SP, f(x) = 0
when x = 1
t.p. at A is: b)
Graph is
moved 2 units to the left, and 4 units up
t.p.’s are: c)
For 3 real roots, line y = k has to cut graph at 3 points
from the graph, k  4

50. a) Find
b) If find in its simplest form. a) b)

51. a) b) c) Functions f and g are defined on the set of real numbers by
a) Find formulae for
i) ii)
b) The function h is defined by
Show that and sketch the graph of h.
c) Find the area enclosed between this graph and the x-axis. a) b) c)
Graph cuts x axis at 0 and 1
Now evaluate
Area

52. The functions f and g are defined on a suitable domain by
a) Find an expression for b) Factorise a) b)
Difference of 2 squares
Simplify