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.