Week 6 and 7 – SS1 Second Term Further Mathematics Notes

WEEK SIX                                                  DATE……………

Revision of half term work

WEEK SEVEN                                             DATE……………

CONTENT:

Composite Mapping and Inverse Mapping

 COMPOSITE MAPPING:
A mapping is composite when the co- domain of the first mapping is the domain of the second mapping.
Consider the mapping f;X→ Z and g: Z→Y

  X Z Yf     g


The mapping f takes an element in X and produces an image in Z, and then the mapping g takes an element in Z and produces an image in Y. It can be denoted by gf or gof.

 Example 1. The mappings f and g are defined by the diagram below:

  F g

 
 
 
 
 
 
 Determine
(a) f(-3) +f(4) (b) f(2) +g(-5) (c) g[f(-3)] (d) g[(-3)] +g[f(4)]

 SOLUTION:

1. F(-3) + F(4) = -5 + 9 = 4
2. F(2) + F(-5) = 5 +(-10) = -5
3. G[f(-3)] = g(-5) = -10
4. G[f(-3)] + g[f(4)] = g(-5) + g(9) = -10 + 18 = 8

 Example 2: The functions f and g on the set of real numbers are defined by f(x) = 3x-1 and g(x) = 5x+2 respectively. Find (a) F [g(x)] (b) g [f(x)] (c) 2f(x) – g(x)
SOLUTION:

1. f[g(x)] = f(5x+2), 5x+2 will represent x in f(x) f ( 5x +2) = 3 (5x+2) – 1

= 15x +5

1. g [f(x)] = g( 3x-1)

= g(3x-1) = 5(3x-1) + 2
= 15x -5 +2
= 15x-3

1. 2f(x) – g(x) = 2(3x-1) – (5x+2)

= 6x -2 -5x -2
= x-4
INVERSE MAPPING:
A function has an inverse if it’s both one- one and onto. Consider the function f(x) = x-3 on the set p={ -1, 5, 9} into set Q ={ -2,1,3 }
A F B

 
           
 
 
 If we reverse the function, that is making range of F to be the domain of the inverse function.
Therefore, B g A

 
 
  g represents the inverse function of f i.e. f^-1, to obtain function f^-1, we follow the procedure below: f(x) = x – 3 2 y = x-3
2
Make x the subject of the formula
2y = x-3 x = 2y + 3 Then, f^-1(x) = 2x+3.

 Example: The function f is defined on the ser of real numbers by f(x) = 2x-1, (x≠-2/3)
3x +2
Determine (a) f^-1(x) (b) f^-1 (-2) (c) determine the largest domain of f^-1(x)

 
 SOLUTION:

1. F(x) = 2x-1 3x+2 f^-1(x), y = 2x-1 3x+2

(3x+2) y = 2x-1
3xy +2y = 2x-1 3xy-2x= -1-2y x (3y-2)= -1-2y x = -1-2y 3y -2 f^-1= -1-2x 3x-2

1. f^-1(-2) i.e x= -2 in f^-1(x) f^-1(-2) = -1-2(-2) 3(-2)-2

= -1+4
-6    2
=    3
(c) The largest domain of f^-1(x) is all real values of x except 2/3     8

EVALUATION

1.     Given g(x) = x³ and h(x) = 4x +1

1. find the value of g(2) + h(2)
2. find the value of h[g(2)]
3. find the value of 3g(-1)-4h(-1)

 2.     A function g(x)=√(x-2), x ≥ 2, find g^-1(x) and g^-1(4)

GENERAL EVALUATION

1. Determine the values of p and q if (x -1) and (x + 2) are factors of 2x³ + px² –x+ q
2. If f(x) = 6x³ + 13x² +2x – 5, shows that f(-1)= 0
3. Given that f(x) = x²
   and g(x)= √(x- 2), x≠2 find x² + 2
   1. f^-1(x)
   2. g^-1(x)
   3. F(g(x)
   4. The value of x for which f(g(x) is not undefined.

 READING ASSIGNMENT: Read Mapping, Further Mathematics Project 2, and page 32- 41.

WEEKEND ASSIGNMENT

1. Given that f(x) = x²+ 4x+ 3, for what values of x is f(x) = f(x+1).

A. -11 B -5 C -3
2 2 4

1. Given f(x) = x² -1 and g(x) = 2x+3, determine the formula for gf(x)

A. 2x² +4x+1 B. 2x² +1 C. x+1

1. Given g(x) = xⁿ and g (3) = 81, determine the value of n.

A -4 B 27 C 4

1. Given that the image of x under the mapping f(x) → 3x +2 is -10. What is the value of x.

A -4 B -28 C 0

1. If f is a function defined by f(x) = 2x-3, find ff. A. 4x-6 B 2x+3 C. 4x +6

    THEORY

1. Given the functions f(x) = 3x² –x+5, g(x) = 6x³ + 7×2+7x+15. Simplify, as far as possible, the expressions     (a) 3f(x) – g(x) (b) f(x) g(x) (c) g(x)/f(x)

1. A relation R is defined by g(x) = 2 , x ≠ 2, find g^-1(x).

x-2