2005 PII No 2
Triangles ACD and BCD are right-angled at D with angles p and q
and lengths as shown in the diagram.
A
84
a) Show that the exact value of sin (p + q) is
85
b) Calculate the exact values of:
(i) cos (p + q)
(ii) tan (p + q)
17
C qp
8
10
4
15
D
6
B
3
The Domain and Range of a Function
The domain of a function is the set of values to
which the function can be applied. (ie the values
x can take)
The range of a function is the set of all possible
output values. (ie the y values)
A function is only fully defined if we are given both:
the rule that defines the function, for example f(x) = x – 4.
the domain of the function, for example the set {1, 2, 3, 4}.
More commonly, a function is defined over a continuous interval
rather than a discrete set of values.
Restricting the Domain
When the domain of a function is not given it is assumed to be the
set of all real numbers, R. That is, x  R.
However, there are many rules that would not be functions if they
were applied to the whole of the set R.
Consider f(x) =
1
x
To make f(x) a function we have to
restrict the domain as follows:
y
1
f(x) = x , x  R, x ≠ 0.
0
x
The range of this function is
f(x)  R , f(x) ≠ 0.
Restricted Domains
x  2.
g(x) =
Since we cannot find the square root
of a negative number the function is
not defined for x < 2.
y
0
2
So, to make g(x) a function we have
to restrict the domain as follows:
x
g(x) =
x2 ,x≥2
The range of this function is f(x) ≥ 0.
This means x can
be any real number
greater than or
equal to 2.
Composite Functions
1
f ( x)  2
x 4
g ( x)  x  1
1. Find an expression for h(x) = f(g(x)).
Give your answer as a single fraction.
2. State a suitable domain for h.
Inverse Functions
Notation:
f (x)
f (x)
- function
f -1(x)
- inverse function
f (x)
x
f -1(x)
Domain
Range
f ( f -1(x)) = x
f -1(f (x)) = x
1. f (x) = x - 5
f
-1 (x)
2. g (x) = 3x
= x+5
4. k (x) = x3
k
-1 (x)
=
3
x
g -1 (x) = 1/3 x
3. h (x) = 3x + 2
h
-1 (x)
=
x2
3
Graphs of Inverse Functions
To find the graph of an inverse function, reflect the graph of the function in the line
y=x
y
y = f (x)
y = f (x)
y=f
y
-1(x)
y=f
x
y=x
x
y=x
-1(x)
Graphs of the form y = ax
y
y = ax
y
y = ax
(0 < a < 1)
(a > 1)
1
1
x
Growth Function
x
Decay Function
Logarithmic Function:
The inverse function of f (x) = ax is called the logarithmic Function to the base a
and is written as log a x
f (x) = a x
f -1(x) = log a x
f (x) = log a x
f -1(x) = a x
f (x) = 3x
y=ax
y
f -1(x) = log 3 x
y = loga x
1
f (x) = ½ x
1
y=x
x
f -1(x) = log ½ x
Graphs of Related Exponential Functions
This sketch shows y = ax + b.
y
(3, 28)
2
0
x
a) Find the values of a and b.
b) Sketch y = loga(x – b), the
inverse of y = ax + b.
Given (0,2). Sub x = 0 and y = 2 into
x
y=a +b
2 = a0 + b
2=1+b
b=1
Graphs of Related Exponential Functions
28 = a3 + b
28 = a3 + 1
27 = a3
a=3
b) To sketch the inverse of y = ax + b we need to reflect the
curve in the line y = x. To reflect a point in y = x ‘swap’ the coordinates.
(3, 28) gives
y=
ax
+b
y = loga(x – b),
(0, 2)
(2, 0)
(3, 28)
28, 3)
y
(28, 3)
0
2
x
Graphs of Related Logarithmic Functions
y
y = log5 x
(5, 1)
0
1
This is the graph of y = log5x.
Sketch the graph of y = log5x + 1.
Sketch the graph of y = log5 (x – 4).
x
Related Graphs
Given y = f (x) , certain transformations can take place which will effect the graph
1.
y = f (x) + c
Graph moves up or
down by c units
y
(2, 6)
(2, 4)
y = f (x) + 2
2
x
0
(2, 0)
y = f (x)
-4
y = f (x) - 4
2.
y = f (x + c )
Graph moves c units to left if c +ve
c units to right if c -ve
y
y = f (x + 2)
-2
(0, 4)
0
(2, 4)
(4, 4)
y = f (x - 2)
x
2
y = f (x)
Sketch y = f (x + 1) - 2
y
(3, 2)
x
0
(-1, -2)
2
(1, -2)
y = f (x)
-4
y = f (x + 1) - 2
3.
y = - f (x)
y
Reflects the graph
in the x-axis
(2, 4)
y = - f (x)
x
0
y = f (x)
(2, -4)
4.
y = f (- x)
Reflects the graph
in the y-axis
y = f (- x)
y
(-2, 4)
(2, 4)
x
-5
0
5
y = f (x)
2004 Paper 1 Q4
The diagram shows the graph of y = g(x).
y
(b, 3)
(a) Sketch the graph of y = -g(x).
(0, 1)
(b) On the same diagram, sketch
the graph of y = 3 - g(x).
O
(a, -2)
(a, 5) y
y =3 -g(x)
b) y = 3 – g (x)
a)
y = -g(x)
(a, 2)
(0, 2)
O (b, 0)
(0, -1)
(b, -3)
x
y = g(x)
y = – g (x) + 3
2
x
2
5.
y = k (f (x))
y
(4, 6)
Multiplies every y
co-ord by k
y = 3 f (x)
(4, 2)
x
(2, -1)
(2, -3)
y = f (x)
6.
y = f (kx)
Divides every x co-ord by k
y
(1, 4)(2, 4)
0
3
y = f (2x )
x
6
y = f (x)