Question

Consider the graph of a real-valued continuous function $f(x)$ defined on R (the set of all real numbers) as shown below.

Find the number of real solutions of the equation $f (f(x)) = 4$

Answer 1

Put $f (x) = t \Rightarrow f (t) = 4 \Rightarrow t = - 2, 0, 4$
Hence $f (x) = - 2\Rightarrow 2$ values of $x$.
or$ f (x) = 0 \Rightarrow 2$ values of $x$.
or $f (x) = 4 \Rightarrow 3$ values of $x$.
Hence, the number of real solutions of the equation$ f (f(x)) = 4 $are $7$.