Finding roots of a specific Function

Question

How can I calculate the roots of this equation? What about in MATLAB?

x2=N(N(x2))

The plot of the function is in below:

Answer 1

If I understand correctly, $N$ is the function defined piecewise by

$$ N(x) = \begin{cases} 5 & x \leq 2, \\ 9 - 2x & 2 < x < 4, \\ 1 & x \geq 4. \end{cases} $$

We can then calculate what $N(N(x))$ is.

• If $x \leq 2$ then $N(x) = 5$, so $N(N(x)) = N(5) = 1$.

• If $x \geq 4$ then $N(x) = 1$, so $N(N(x)) = N(1) = 5$.

• If $2 < x < 4$ then $N(x) = 9-2x$. Since we're plugging this back into $N$, we want to know when $2 < 9-2x < 4$. Subtracting $9$ from this gives $-7 < -2x < -5$, and dividing by $-2$ gives $5/2 < x < 7/2$.

  In the range $5/2 < x < 7/2$ we thus have

  $$ N(9-2x) = 9 - 2(9-2x) = 4x - 9. $$

  In the range $x \leq 5/2$ we have $9-2x \geq 4$, so that here $N(9-2x) = 1$.

  In the range $x \geq 7/2$ we have $9-2x \leq 2$, so that here $N(9-2x) = 5$.

  In summary,

  $$ N(9-2x) = \begin{cases} 1 & x \leq 5/2, \\ 4x - 9 & 5/2 < x < 7/2, \\ 5 & x \geq 7/2. \end{cases} $$

Combining these three bullet points, we conclude that

$$ N(N(x)) = \begin{cases} 1 & x \leq 5/2, \\ 4x - 9 & 5/2 < x < 7/2, \\ 5 & x \geq 7/2. \end{cases} $$

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Now, you are interested in determining which values of $x$ satisfy

$$ x = N(N(x)). \tag{1} $$

Since $N(N(x)) = 1$ for $x \leq 5/2$, the only solution to equation $(1)$ with $x \leq 5/2$ is $x = 1$.

If $5/2 < x < 7/2$ then $N(N(x)) = 4x - 9$, and if we set

$$ x = 4x-9 $$

then we get $x = 3$. This falls in the range $5/2 < x < 7/2$, so this is a true solution to equation $(1)$.

If $x \geq 7/2$ then $N(N(x)) = 5$, so the only solution to equation $(1)$ in this interval is $x=5$.

In total, the solutions to equation $(1)$ are $x=1$, $x=3$, and $x=5$.

Here is a plot of $N(N(x))$ in blue versus $x$ in orange.

This was created in Mathematica using the code

n[x_] := Piecewise[{{5, x < 2}, {9 - 2 x, 2 < x < 4}, {1, x > 4}}];
Plot[{n[n[x]], x}, {x, 0, 6}, PlotPoints -> 70]