Advance Mathematics

Advance
Mathematics

Table of Contents
INTRODUCTION...........................................................................................................................1
Question-1........................................................................................................................................1
Question-2........................................................................................................................................2
Question -3.......................................................................................................................................3
Question 4 .......................................................................................................................................6

INTRODUCTION
Question-1
A) Give the function f(x) = 2√x+4
I) Find F(x)=0
f(0)=2√x+4
=2√0+4
=2*2
=4
II) f(x) = 9x^2
F(9x^2) = 2√x+4
=2√9x^2+4
=2√3x+2
III) Find domain and range for this function
f(x) = 2√x+4
Domain= x≥ -4
Range= f≥ 0 (all non-negative real numbers)
B) Show that f(x) =2x+3 and g(x)=x-3/2 are inverse
Given function where
f(x)=2x+3
let the y=2x+3
let the inverse function g(y)=x
Find the x in terms of y
Therefore,
2x=y-3
x=(y-3)/2
Therefore,
g(y)=(y-3)/2
y is the dummy variable of this function
Hence g(y) =(x-3)/2 which is an inverse function of f(x)=2x+3
1

c) The function f is defined by :
I) f(x)= 3(x+1)/(2x^2+7x-4) – (1/(x+4))
=3(x+1)/ (2x^2+8x-x-4) – (1/(x+4))
=3(x+1)/ (2x(x+4)-1(x+4)) – (1/(x+4))
=3(x+1)/ ((x+4) (2x+1))- (1/(x+4))
= (1/(x+4))( (3x+3-2x+1)/ (2x-1))
= (1/(x+4)) ((x+4)/ (2x+1))
=1/(2x+1)
Hence proved.
ii) Find f^-1 (x)
f^-1 (x)= 1/(2x+1)
= 2x-1
= x= 1/2
iii) Find the domain of f^-1
f^-1= (2x-1)
where x= ½ so the domain of this function are the all real numbers.
iv) Given that g(x) = In (x+1)
Find the solution of g(x) =1/7 give your answer in term of e.
=(1/7+1)
= 0.14+1=1.14
Question-2
The function f has domain -2≤x≤6 and is a linear from (-2,10) to (2,0) and from (2,0)
to (6,4). A sketch of the graph of y=f(x) is shown in figure 1.
I) write down the range of (f)
0≤F (X) ≤,0
II) Find ff(0)
The function is defined by g:x= (4-3x)/(5-x)
F(0)=5
F(5)=3
III) Find g^-1(x)
Y=4+3X/5-X
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