From the graph find the number of solutions.

Question

The figure below shows the function $f(x)$ .

How many solutions does the equation $f(f(x))=15$ have ?

$a.)\ 5 \\ b.)\ 6 \\ c.)\ 7 \\ d.)\ 8 \\ \color{green}{e.) \ \text{cannot be determined from the graph}}$

From figure $f(x)=15$ occurs at $x\approx \{4,12\}$ and

$f(x)=4$ occurs at $4$ points and $f(x)=12$ occurs at $3$ points.

so i concluded answer is option $c.$

I look for a short and simple way.

I have studied maths up to $12$th grade.

Answer 1

There are two things that could go wrong with your reasoning:

1. If $f$ is defined for $x < -10$ or for $x > 13$, you have no idea how it behaves.
2. You can't be sure that there is exactly one value of $x$ near $4$ so that $f(x)=15$. It could be the case that there is none, or one, or two. This depends on whether the value of the local maximum of $f(x)$ when $3<x<5$ is (slightly) less than, equal to, or (slightly) greater than 15.

Answer 2

I don't think that it is obvious that there is a real number $\alpha$ such that $3\lt \alpha\lt 5$ and $f(\alpha)=15$.