Which one of the following is correct in respect of the function f: R → R⁺ defined as f(x) = |x + 1|?

Question

Question

This question was previously asked in

NDA (Held On: 22 April 2018) Maths Previous Year paper

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Answer 1

Calculation:

Given that,

⇒ f(x) = |x + 1|

first option

⇒ f(x²) = |x² + 1|

⇒ f(x)² = |x + 1|²

so we can say that

⇒ f(x²) = |f(x)|²

⇒ |x + 1|² ≠ |x² + 1|

First option not correct.

Second option,

⇒ f(|x|) = |f(x)|

⇒ f(|x|) = ||x| + 1|

⇒ |f(x)| = ||x + 1|| = |x + 1|

f(|x|) ≠ |f(x)| because ||x| + 1| ≠ |x + 1| for real values of x.

Third option,

⇒ f(x + y) = f(X) + f(y)

⇒ f(x + y) = |(x + y) + 1|

⇒ f(y) = |y + 1|

so,

⇒ |(x + y) + 1| ≠ |x + 1| + |y + 1|

So f(x + y) ≠ f(x) + f(y)