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F(x)=4^x/4^x+2,then f(x)+f(1-x) is equal to A)0 b)-1 c)1 d)4

F(x)=4^x/4^x+2,then f(x)+f(1-x) is equal to
A)0 b)-1 c)1 d)4

Grade:12

2 Answers

Saurabh Koranglekar
askIITians Faculty 10341 Points
2 years ago
576-209_1.jpg
Akshat Agrawal
24 Points
2 years ago
Let the Equation (1) be -      ​f(x) + f(1-x) = \frac{4^{x}}{4^{x} + 2} + \frac{4^{1-x}}{4^{1-x} + 2} = \frac{4^{x}(4^{1-x} + 2) + 4^{1-x}(4^{x} + 2)}{(4^{x} + 2)(4^{1-x} + 2)}
 
Now we can write ,   4^{1-x}(4^{x} + 2) = 4 + 2*4^{1-x} = 2(2 + 4^{1-x}) = 2(4^{1-x} + 2)    (all i did is to multiply the numbers and rearrange them in a specific order).Let this be equation (2)
 
So we have equation (1) (using equation (2)) as - 
 f(x) + f(1-x) = \frac{4^{x}(4^{1-x} + 2) + 4^{1-x}(4^{x} + 2)}{(4^{x} + 2)(4^{1-x} + 2)} = \frac{4^{x}(4^{1-x} + 2) + 2(4^{1-x} + 2)}{(4^{x} + 2)(4^{1-x} + 2)} \\\\ = \frac{(4^{1-x} + 2)(4^{x} + 2)}{(4^{x} + 2)(4^{1-x} + 2)} = 1
 
This gives us the answer as 1. So the answer is option (c)

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