Question

The graph of a function $y=f(x)$ is shown in the figure

Identify which of the following statement is/are correct?

• A. Domain of $f(f(x))$ is $[-4,3]$.
• B. Domain of $f(f(x))$ is $\left[-3,\frac{13}{5}\right]$.
• C. The equation of $f(f(x))=1$ has exactly 7 solutions.
• D. The equation of $f(f(x))=1$ has exactly 8 solutions.

Answer 1

Answer: (B), (C)

Domain of $y=f(f(x))$
Let $f(x)=t\Rightarrow t\in [0,5]$ $y=f(t)$
But domain of $f$ is $[-4,3]$
$\therefore t\in [-4,3]\Rightarrow f(x)\in [-4,3]\Rightarrow x\in \left[-3,\frac{13}{5}\right]$
Number of solution(s) of the equation $f(f(x))=1$
$\underset{\downarrow }{f(t)=1},f(x)=t$
$t=-1,0,\frac{3}{2},\frac{11}{5}$
Now, $f(x)=-1\Rightarrow$ No solution
$f(x)=0\Rightarrow 2$ solutions
$f(x)=\frac{3}{2}\Rightarrow 3$ solutions
$f(x)=\frac{11}{5}\Rightarrow 2\text{ solutions }$
Hence, $f(f(x))=1$ has exactly 7 solutions.