Let f : R - {3} → R - {1} be defined by f(x) = (x-2)/(x-3). Let g : R→R be given as g(x) = 2x - 3. Then, the sum of all the values of x for which f?¹(x) + g?¹(x) = 13/2 is equal to :
Let f : R - {3} → R - {1} be defined by f(x) = (x-2)/(x-3). Let g : R→R be given as g(x) = 2x - 3. Then, the sum of all the values of x for which f?¹(x) + g?¹(x) = 13/2 is equal to :
Option 1 -
2
Option 2 -
7
Option 3 -
5
Option 4 -
3
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1 Answer
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Correct Option - 3
Detailed Solution:Given functions f (x) = (x-2)/ (x-3) and g (x) = 2x-3.
First, find the inverse functions f? ¹ (x) and g? ¹ (x).
For f? ¹ (x): y = (x-2)/ (x-3) ⇒ y (x-3) = x-2 ⇒ xy - 3y = x-2 ⇒ xy-x = 3y-2 ⇒ x (y-1) = 3y-2 ⇒ x = (3y-2)/ (y-1). So, f? ¹ (y) = (3y-2)/ (y-1).
For g? ¹ (x): y = 2x-3 ⇒ y+3 = 2x ⇒ x = (y+3)/2. So, g? ¹ (y) = (y+3)/2.
We are given f? ¹ (x) + g? ¹ (x) = 13/2.
(3x-2)/ (x-1) + (x+3)/2 = 13/2.
Multiply by 2 (x-1): 2 (3x-2) + (x+3) (x-1) = 13 (x-1).
6x - 4 + x² + 2x - 3 = 13x - 13.
x² + 8x - 7 = 13x - 13.
x² - 5x + 6 = 0.
(x-2) (x-3) = 0.
The possible...more
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