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

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 $f(f(x))=4.$

I found $f(x)=2x+8,\text{when} x\leq -2$
$f(x)=-2x,\text{when} -2<x\leq-1$
$f(x)=2x+4,\text{when} -1<x\leq0$
$f(x)=4-x,\text{when} 0<x\leq 2$
$f(x)=x,\text{when} 2<x\leq 4$
$f(x)=-4x+20,\text{when} x>4$

I tried to find the composition of the piecewise defined function.But i could not find $f(f(x))$

Please help me.

Answer 1

Hints:

• If $f(y)=4$, what can $y$ be?
• If $f(z)=y$ where $f(y)=4$, what can $z$ be?
• If $f(f(x))=4$, how many possible values of $x$ are there?

Reading off the graph may help.

Answer 2

Looking at the graph treating $u$ as a dummy variable we see:

$f(u)=4$ for $u=-2,0,4$

Therefore we need to find values of $x$ s.t:

$u=f(x)=-2,0,4$

This happens $7$ times.