If a function $ f: R \rightarrow R $ is defined as $ f(x)=\int \frac{x^{8}+4}{x^{4}-2 x^{2}+2} d x $ and $ f(0)=1 $, then which of the following is correct?

Question

If a function $ f: R \rightarrow R $ is defined as $ f(x)=\int \frac{x^{8}+4}{x^{4}-2 x^{2}+2} d x $ and $ f(0)=1 $, then which of the following is correct?, $ f(x) $ is an even function, $ f(x) $ is an onto function, $ f(x) $ is an odd function, $ f(x) $ is many one function

MARKS App · Accepted Answer

We have, f x = ∫ x 8 + 4 x 4 - 2 x 2 + 2 d x Now, f x = ∫ x 8 + 4 + 4 x 4 - 4 x 4 x 4 - 2 x 2 + 2 d x = ∫ x 4 + 2 2 - 2 x 2 2 x 4 - 2 x 2 + 2 d x = ∫ x 4 + 2 x 2 + 2 x 4 - 2 x 2 + 2 x 4 - 2 x 2 + 2 d x Therefore, f x = x 5 5 + 2 x 3 3 + 2 x + C ⇒ f 0 = 0 + 0 + 0 + C = 1 ⇒ C = 1 ⇒ f x = x 5 5 + 2 x 3 3 + 2 x + 1 Range of f x is R , So, f ( x ) is an onto function f - x ≠ f x , So, f x is not even f - x ≠ - f x , So, f x is not odd f ' x > 0 ∀ x ∈ R , So, f ( x ) is one-one

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