Question

If f(x) + f(x – 1) = 2 ∀ x∈R, g(x) = f(x) – 1, if g(1) = 3 then (g(1))2 – g(–1) is equal to


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Answer 1

Given info :  f(x) + f(x – 1) = 2 ∀ x∈R , g(x) = f(x) – 1.

To find :  if g(1) = 3, g(1)² - g(-1) is equal to ..

solution : g(x) = f(x) - 1

g(1) = f(1) - 1 [ putting x = 1 ]

⇒ 3 = f(1) - 1

⇒ f(1) = 4 ...(1)

also f(x) + f(x - 1) = 2

f(1) + f(0) = 2 [ putting x = 1 ]

from equation (1),

4 + f(0) = 2

⇒ f(0) = -2 ...(2)

again putting x = 0

f(0) + (-1) = 2

⇒ -2 + f(-1) = 2 [ from eq (1) ]

⇒ f(-1) = -4 ...(3)

now, g(x) = f(x) - 1

g(-1) = f(-1) - 1 [ putting x = -1 ]

= -4 -1 = -5 [ from equation (3), ]

therefore the value of g(1)² - g(-1) = 3² - (-5) = 9 + 5 = 14