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

Correct answer: D. None of the above Given, f(x)=|x+1| . Let us check all options. (a) aligned & f(x)^2=|x^2+1| & (f(x))^2=(x+1)^2 & So, f(x)^2 ≠(f(x))^2 aligned (b) aligned & f(|x|)=\| x|+1| & |f(x)|=|x+1||=|x+1| & f(x) ≠|f(x)| aligned (c) aligned & f(x+y)=|x+y+1| & f(x)+f(y)=|x+1|+|y+1| & f(x+y) ≠ f(x+f(y)) aligned

NDA 2018

Exam: NDA
Year: 2018
Subject: Mathematics
Chapter: Functions

NDA 2018 · Mathematics · Functions

Which one of the following is correct in respect of the function f:...

Which one of the following is correct in respect of the function

\mathrm{f}: \mathbb{R} \rightarrow \mathbbR^{+}

defined as

\mathrm{f}(\mathrm{x})=|\mathrm{x}+1|

Options

A $\mathrm{f}\left(\mathrm{x}^2\right)=[\mathrm{f}(\mathrm{x})]^2$
B $\mathrm{f}(|\mathrm{x}|)=|\mathrm{f}(\mathrm{x})|$
C $f(x+y)=f(x)+f(y)$
DNone of the above

Correct answer

D. None of the above

Step-by-step solution

Given,

\mathrm{f}(\mathrm{x})=|\mathrm{x}+1|

. Let us check all options. (a)

\begin{aligned} & \mathrm{f}(\mathrm{x})^2=\left|\mathrm{x}^2+1\right| \\ & (\mathrm{f}(\mathrm{x}))^2=(\mathrm{x}+1)^2 \\ & \text{ So, } \mathrm{f}(\mathrm{x})^2 \neq(\mathrm{f}(\mathrm{x}))^2 \end{aligned}

(b)

\begin{aligned} & \mathrm{f}(|\mathrm{x}|)=\| \mathrm{x}|+1| \\ & |\mathrm{f}(\mathrm{x})|=|\mathrm{x}+1||=|\mathrm{x}+1| \\ & \mathrm{f}(\mathrm{x}) \neq|\mathrm{f}(\mathrm{x})| \end{aligned}

(c)

\begin{aligned} & \mathrm{f}(\mathrm{x}+\mathrm{y})=|\mathrm{x}+\mathrm{y}+1| \\ & \mathrm{f}(\mathrm{x})+\mathrm{f}(\mathrm{y})=|\mathrm{x}+1|+|\mathrm{y}+1| \\ & \mathrm{f}(\mathrm{x}+\mathrm{y}) \neq \mathrm{f}(\mathrm{x}+\mathrm{f}(\mathrm{y})) \end{aligned}

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