Let f:[-6,6] arrow IR defined by f(x)=x^2-3 . Consider the following: 1. (fofof)(-1)=(fofof)(1) 2. (fofof)(-1)-4(fofof)(1)=(f^ ^ f)(0) Which of the above is correct?

Correct answer: C. Both 1 and 2 aligned & f(x)=x^2-3 & fof (x)=f(f(x))=f(x^2-3)=(2^2-3)^2-3 & =2^4-6 x^2+9-3=x^4-6 x^2+6 & fof (0)=0-0+6=6 & f(f(f(x)))=f(x^4-6 x^2+6)=(x^4-6 x^2+6)^2-3 & fofof (1)=((-1)^4-6(-1)^2+6)^2-3 & =(1-6+6)-3=-2 & fofof (1)=(1^4-6(1)^2+6)^2-3 & =(1-6+6)-3=-2 aligned

NDA 2017

Exam: NDA
Year: 2017
Subject: Mathematics
Chapter: Functions

NDA 2017 · Mathematics · Functions

Let f:[-6,6] arrow IR defined by f(x)=x^2-3 . Consider the...

Let

f:[-6,6] \rightarrow IR

defined by

f(x)=x^2-3

. Consider the following: 1.

\quad\left(fofof\right)(-1)=\left(fofof\right)(1)

\left(fofof\right)(-1)-4\left(fofof\right)(1)=\left(f^{{}^{\circ}} f\right)(0)

Which of the above is correct?

Options

A1 only
B2 only
CBoth 1 and 2
DNeither 1 nor 2

Correct answer

C. Both 1 and 2

Step-by-step solution

\begin{aligned} & \mathrm{f}(\mathrm{x})=\mathrm{x}^2-3 \\ & \text{fof }(\mathrm{x})=\mathrm{f}(\mathrm{f}(\mathrm{x}))=\mathrm{f}\left(\mathrm{x}^2-3\right)=\left(2^2-3\right)^2-3 \\ & =2^4-6 \mathrm{x}^2+9-3=\mathrm{x}^4-6 \mathrm{x}^2+6 \\ & \text{fof }(0)=0-0+6=6 \\ & \mathrm{f}(\mathrm{f}(\mathrm{f}(\mathrm{x})))=\mathrm{f}\left(\mathrm{x}^4-6 \mathrm{x}^2+6\right)=\left(\mathrm{x}^4-6 \mathrm{x}^2+6\right)^2-3 \\ & \text{fofof }(1)=\left((-1)^4-6(-1)^2+6\right)^2-3 \\ & =(1-6+6)-3=-2 \\ & \text{fofof }(1)=\left(1^4-6(1)^2+6\right)^2-3 \\ & =(1-6+6)-3=-2\end{aligned}

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