If f(x) 2f(1-x)=(x^2) 1,for all x€ R ,then the range of f is ?

Question

Answer

Range of f(x) in the given equation

To find the range of f(x) in the given equation f(x) 2f(1-x) = x^2 1, we can start by simplifying the equation and manipulating it to express f(x) in terms of x.

Solving the equation
Given f(x) 2f(1-x) = x^2 1, let's substitute y = 1-x:
f(1-y) 2f(y) = (1-y)^2 1
f(y) 2f(1-y) = y^2 1

Comparing the two equations
By comparing the two equations, we can see that f(y) = y^2 1 is the same as f(x) = x^2 1. Therefore, the range of f(x) is all real numbers.

Range of f(x)
Since f(x) = x^2 1 for all x, the range of f(x) is all real numbers.

In conclusion, the range of the function f(x) in the given equation is all real numbers. This is because the function f(x) takes on all possible values as x varies over the real numbers, as demonstrated by the equation f(x) = x^2 1.

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