1.1 Relations
1.1.1 Relation – Domain and Codomain
1.1.2 Types of Relation
1) The relation which maps Set A = {6, 12, 18, 23} to set B = {1, 2, 3, 4} is “remainder when divided
by 5 is”.
a) I) arrow diagram
II) graph
III) ordered pairs
b) Identify the:
Domain
=
Co-domain =
Range
=
Image of 12 =
image of 23 =
object of 1 =
object of 3 =
2) Complete the table below.
Relations
i)
2
5
3
6
4
7
1
2
3
ii) {(3, 7), (4, 12),(5, 4),(6,
12)}
Domain
Co-domain
Object
Image
Range
1.2 Functions
1) Given the function f : x → 2x + 5 , find
i)
f(3)
ii) f(-4)
2) Given the function g : x → x2 – 3x + 1
i)
g(2)
ii) g(-1)
3) Given that f : x → 3x – 2, find
i) image of -2
ii) the value of x if f(x) = 10
4) Given that f : x → 5x + 3 , find the value of x if f(x) = -7
5) Given that f:x → 3x +7, find the value of x if f(x) = 4
6) I) Given that g : y → y + 2 , find the value of y if g(y) = 2y – 3.
II) Given that g : y → 3y – 2 , find the value of y if g(y) = 2y + 4
7) I) Given that f: x → 7x – 3 , g : x → 4x + 15, find the value of p if f(p) = g(p).
II) Given that f : x → 2x – 10 , g : x → 4x + 2 , find the value of x if f(x) = g(x)
8) The arrow diagram shows the function 𝑓 ∶ 𝑥 →
x
15
𝑎𝑥+𝑏
,𝑥 ≠
−𝑏
𝑎
15
𝑓 ∶ 𝑥 → 𝑎𝑥+𝑏
6
3
1
1
a) Find the values of a and b.
b) State the value of x if f undefined.
3x+k
13) A function f is defined by 𝑓 ∶ 𝑥 → 2𝑥−4 for all values of x, except x = p and k is a constant.
a) State the value of p.
b) Given that the value 5 is mapped onto itself under f, find
I)
The value of k
II)
Another value of x that is mapped onto itself.
14) Sketch the graph of each of the following absolute value functions.
a) f(x) = |2x - 8| for the domain 0 ≤ x ≤ 10
b) f(x) = |3 – 2x| for the domain -3 ≤ x ≤ 4
1.3 Composite Functions
Noted: f2(x) = ff(x) , but not f2(x) = [f(x)]2
1) Given that f : x → 3x – 4 and g : x → 2x , find fg(3).
2) Given that f : x → 3 – 2x and g : x → x2 , find gf(4).
3) Given that f : x → 3x -4 and g : x → 2x , find fg(x)
4) Given that f : x → 2x – 5 and g : x → 5x , find the composite function gf
5) Given f(x) = 4x – 1 and g(x) = 3x. Determine the following composite functions.
a) fg
b) gf
c) (gf)2
d) gf2
2
6) Given that 𝑓 ∶ 𝑥 → 𝑥 , x ≠ 0 and g : x → 2x – 3 , find the value of x for which
a) f = g
b) f2 = g2
7) Given a function f and a composite function fg. Determine the function g.
a)f : x → x – 2, fg : x → x2 + 3
b) f : x → 2x + 4 , fg : x → 8 – 2x
c) f : x → 5x + 1 , fg : x → 10x2 -6
d) 𝑓: 𝑥 →
1
, 𝑥 ≠ 2 , 𝑓𝑔: 𝑥
𝑥−2
→
1
,𝑥 ≠ 5
𝑥−5
8) Given a function f and a composite function gf. Determine the function g.
a) f : x → x – 3 , gf : x → x2 + 2
b) f : x → 2x , gf : x → 6x – 8
c) f : x → x + 1 , gf : x → x2 – x + 3
d) f : x → x – 2 , gf : x → x2 + 4x – 5
9) Given f(x) = px + 2 , g(x) = 4x + c and fg(x) = 12x – 1 , find the value of p and c.
10) Given 𝑓(𝑥) = 2𝑥 + ℎ, 𝑔(𝑥) =
𝑘
5
, 𝑥 ≠ 1 𝑎𝑛𝑑 𝑔𝑓(𝑥) =
, 𝑥 ≠ 2, find the value of h and k.
𝑥−1
2𝑥−4
11) Given that f : x → (x + 1)2 and g : x → x – 2. Find
(a) The composite functions fg and gf.
(b) The value of x if fg = gf.
(c) The value of x if fg = g
12) Given f : x → ax + b where a > 0 and f2 : x → 9x – 8 . Find the value of a and b.
1.4 Inverse Functions
1) Determine the inverse function for each of the following.
a) f : x → 2x – 3
b) h : x → 6 – 3x
c) 𝑓: 𝑥 →
5x+4
3
5
e) 𝑓: 𝑥 → 𝑥−2 , 𝑥 ≠ 2
d) 𝑓: 𝑥 →
(4−x)
5
1
2
f)𝑓: 𝑥 → 3𝑥+2 , 𝑥 ≠ − 3
2) Given that f : x → 3x – 5 , find the value of f-1 (7).
9𝑥+2
3) Given that 𝑓(𝑥) = 5𝑥−3 , 𝑥 ≠ 𝑘
(a) State the value of k
(b) Find f-1(2)
4) Given the function f : x → 3x + c and its inverse function f-1 : x → mx + 4/3 , find the value of m
and c.
5) Given the function h : x → 4x + m and its inverse function h-1 : x → 2kx + 5/8 , find the value of m
and of k.
hx+k
6) Given the function 𝑓: 𝑥 → 𝑥−2 , 𝑥 ≠ 2 and its inverse function f-1 : x →
(2𝑥−5)
𝑥−3
, 𝑥 ≠ 3, find the
value of h and k.
𝑚𝑥
7) Given the inverse function f-1 : x → 𝑥+1 , 𝑥 ≠ −1 and f(10) = 2, find the value of m.
𝑥+3
8) Given that g-1 : x → 2𝑥 , 𝑥 ≠ 0 , find g(x).
Assessment: Functions
𝑏
𝑎𝑥 + 𝑥
1. x
3
14
1
10
Diagram above shows an arrow diagram
representing part of the mapping of x by
𝑏
the function f : x →𝑎𝑥 + 𝑥 , 𝑥 ≠ 0 , Find
(a) The value of a and b
(b) The image of 6 under this mapping
(c) The object which are mapped to 11
2) Given that f : x → 2x + 1 and
5
g : x → 𝑥−2 , 𝑥 ≠ 2 , find
(a) fg
(b) f2
(c) f-1
(d) (fg)-1
3) The inverse of f(x) = 5x – n is given by
4
5
f-1(x) = 𝑚𝑥 + . Find m and n.
4) Given that g(x) = x + 3 and
fg(x) = x2 + 5x – 6, find f(x).
5) Given that f : x → ax + b , a > 0, and
f2 : x → 9x – 8 , find the value of a and b.
2
6) Given the function f : x →𝑥−3 , 𝑥 ≠ 𝑘. Find
a) The value of k
b) f-1 (x)
7) Given f : x → x2 – 2 and g : x → x + 5. Find
a) fg(x)
b) the value of gf(-1)
8) Given the function f(x) = 3x + 4 , find the value of x
a) If f2(x) = f(-x)
b) When x is mapped onto itself
9) Given the function f : x → 2x + 3 , find the
a) fg : x → 2x2 + 3
b) gf : x → 2x – 2
10) Given the functions f : x → x + 3 and
g : x → a + bx2. If gf : x → 6x2 + 36x +56, find
a) The values of a and b
b) The value of gf-1(2)
11) Given g(x) = ax + b and g2(x) = 16x – 25. Find
a) The value of a and b
b) The value of x if 2g(x) = g(x+3) by taking
the positive value of a.
12) Given the inverse function f-1(x) =
2𝑥−3
, find
2
the value of
a) f(4)
b) k if f-1(2k) = – k – 3
6
13) Given f : x → 3x – 4 and g : x → 𝑥 , x≠ 0. Find
a) f-1g(x)
b) gf-1(2)
14) Given that f(x) = |3x - 9|, find
a) f(1)
b) the values of objects that have the image 6
2𝑥−4
and fg : x → 3x + 2 , find the
3
15) Given f-1 : x →
function
a) f(x)
b) g(x)
16)
P = {1 , 3 , 5}
Q = {2 , 4 , 6 , 8}
Based on the above information, the relation between
P and Q is defined by the set of ordered pairs {(1 , 2),
(1, 4), (3, 6), (5, 8)}. State
a) The images of object 1
b) The object of 6
17) Given that f : x → 3x – 1 and g : x → x2 + 5x + 6 . find
a) f-1(2)
b) gf(x)