please solve step by step.. tq Question I (a) The functions f(x),g (x)and h(x)ar

Question

please solve step by step.. tq

Question I (a) The functions f(x),g (x)and h(x)are defined as follows: g(x) = VX h(x) = 2x + 1 xER (i) GFind fg(x) and state its domain and range (ii Find hf (x) and state its domain and range Giii) Find h (x) and state its domain and range [3 marks] [3 marks] [3 marks The figure below shows the graph of y =-r2shifted to four new positions, write an equation for each new graph. (b) [8 marks] (14) (2.0) (c) Find a formula for the inverse of the function. f(x) = 4x-1 2x +3 [3 marks] (ii) f(x) V10-3x 12 marks) 1+e i) 3 marks]

Explanation / Answer

solutionC:

(i)f(x) is mapping x to y,

then the inverse function of f(x) maps y back to x

f(x)=4x-1/2x+3

put f(x)=y

y=4x-1/2x+3

multiply bothsided by 2x+3

y(2x+3)=4x-1/2x+3(2x+3)

y(2x+3)=4x-1

4x-1=2xy+3y

add 1 to both ssids

4x-1+1=2xy+3y+1

4x=2yx+3y+1

x=-(3y+1)/2(-2+y)

substitute y=x

-(3x+1)/2(-2+x)

inverse is -(3x+1)/2(-2+x)

SolutionC(II)

let sqrt(10-3X)=y

square both sides

10-3x=y2

x=-y2 -10/3

substitute y=x

=-x2 -10/3

inverse is

-x2 -10/3

Solutionciii

let

(1+ex)/(1-ex)=y

(1+ex)=y(1-ex)

1+ex =y-yex

subtract 1 from both sides'

1+ex-1=y-yex -1

ex =y-yex -1

ex(y+1)=y-1

ex =y-1/y+1

x=ln(y-1/y+1)

substitte y=x

ln(x-1/x+1)

inverse ln(x-1/x+1)

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