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Question:
Answered & Verified by Expert
If $f:[-6,6] \rightarrow R$ is defined by $f(x)=x^2-3$ for $x \in R$, then
$$
(f \circ f \circ f)(-1)+(f \circ f \circ f)(0)+(f \circ f \circ f)(1)
$$
is equal to
Options:
$$
(f \circ f \circ f)(-1)+(f \circ f \circ f)(0)+(f \circ f \circ f)(1)
$$
is equal to
Solution:
1439 Upvotes
Verified Answer
The correct answer is:
$f(4 \sqrt{2})$
Given, $\quad f(x)=x^2-3$
Now, $\quad f(-1)=(-1)^2-3=-2$
$$
\begin{aligned}
& \Rightarrow \quad f \circ f(-1)=f(-2)=(-2)^2-3=1 \\
& \Rightarrow \quad f \circ f \circ f(-1)=f(1)=1^2-3=-2
\end{aligned}
$$
Now,
$$
f(0)=0^2-3=-3
$$
$$
\begin{aligned}
& \Rightarrow \quad f \circ f(0)=f(-3)=(-3)^2-3=6 \\
& \Rightarrow \quad f \circ f \circ f(0)=f(6)=6^2-3=33
\end{aligned}
$$
Again,
$$
\begin{aligned}
& f(1)=1^2-3=-2 \\
& \Rightarrow \quad f \circ f(1)=f(-2)=(-2)^2-3=1 \\
& \Rightarrow \quad f \circ f \circ f(1)=(1)^2-3=-2 \\
& \therefore f \circ f \circ f(-1)+f \circ f \circ f(0)+f \circ f \circ f(1) \\
& =-2+33-2=29 \\
&
\end{aligned}
$$
Now,
$$
\begin{aligned}
f(4 \sqrt{2}) & =(4 \sqrt{2})^2-3=32-3 \\
& =29
\end{aligned}
$$
Hence, option (a) is correct.
Now, $\quad f(-1)=(-1)^2-3=-2$
$$
\begin{aligned}
& \Rightarrow \quad f \circ f(-1)=f(-2)=(-2)^2-3=1 \\
& \Rightarrow \quad f \circ f \circ f(-1)=f(1)=1^2-3=-2
\end{aligned}
$$
Now,
$$
f(0)=0^2-3=-3
$$
$$
\begin{aligned}
& \Rightarrow \quad f \circ f(0)=f(-3)=(-3)^2-3=6 \\
& \Rightarrow \quad f \circ f \circ f(0)=f(6)=6^2-3=33
\end{aligned}
$$
Again,
$$
\begin{aligned}
& f(1)=1^2-3=-2 \\
& \Rightarrow \quad f \circ f(1)=f(-2)=(-2)^2-3=1 \\
& \Rightarrow \quad f \circ f \circ f(1)=(1)^2-3=-2 \\
& \therefore f \circ f \circ f(-1)+f \circ f \circ f(0)+f \circ f \circ f(1) \\
& =-2+33-2=29 \\
&
\end{aligned}
$$
Now,
$$
\begin{aligned}
f(4 \sqrt{2}) & =(4 \sqrt{2})^2-3=32-3 \\
& =29
\end{aligned}
$$
Hence, option (a) is correct.
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