Finding the Value of $f(84)$ in an Integer Function

[Albert1]

$\text{given } :x \in\mathbb{Z}$

$f(x)= \begin{cases}x-3 & x \geq 1000 \\f\big [f(x+5)\big ]& x<1000 \end{cases} $

$\text{find } :\,\, f(84)$

 

[I like Serena]

$\text{given } :x \in\mathbb{Z}$

$f(x)= \begin{cases}x-3 & x \geq 1000 \\f\big [f(x+5)\big ]& x<1000 \end{cases} $

$\text{find } :\,\, f(84)$

Spoiler

Lemma
$f(x)= \begin{cases}
x-3 & x \geq 1000 \\
997 & x<1000 \text{ and $x$ even} \\
998 & x<1000 \text{ and $x$ odd} \\
\end{cases} $

Proof
Use full induction going down.
Initial condition: we can verify that it is true for any $x \ge 997$.
Induction step: suppose it is true for any $x$ and some $y$ with $y < x$ and $y< 997$.
Then we need to distinguish the cases that $y$ is even or $y$ is odd.
When we fill in what we already have for $f(y)$ it follows that the given formula is also true for $f(y)$, which completes the proof.

Using the lemma we find that $f(84) = 997$.

 

[Albert1]

Spoiler

Lemma
$f(x)= \begin{cases}
x-3 & x \geq 1000 \\
997 & x<1000 \text{ and $x$ even} \\
998 & x<1000 \text{ and $x$ odd} \\
\end{cases} $

Proof
Use full induction going down.
Initial condition: we can verify that it is true for any $x \ge 997$.
Induction step: suppose it is true for any $x$ and some $y$ with $y < x$ and $y< 997$.
Then we need to distinguish the cases that $y$ is even or $y$ is odd.
When we fill in what we already have for $f(y)$ it follows that the given formula is also true for $f(y)$, which completes the proof.

Using the lemma we find that $f(84) = 997$.

I like Serena :very smart induction (Clapping)

 

[kaliprasad]

My approach is different from I like serena

Spoiler

we evaluate f(999) through f(995)

f(999) = 998
f(998) = 997
f(997) = 998
f(996) = 997
f(995) = 998

now f(85)= f^183 (998) notation for f is applied 183

applying f twice gives 998 and so on applying 182 time gives 998 and then once more gives 997 which is the ans

 

[CellCoree]

Based on the given information, we can see that the function $f(x)$ has two different cases depending on the value of $x$. For $x \geq 1000$, the function returns the value of $x-3$, while for $x<1000$, the function returns the result of $f\big [f(x+5)\big ]$.

Since $84<1000$, we will use the second case of the function. This means that $f(84)=f\big [f(84+5)\big ]$.

To find the value of $f(84+5)$, we need to use the first case of the function since $89 \geq 1000$. Therefore, $f(84+5)=84+5-3=86$.

Substituting this back into the original equation, we get $f(84)=f(86)$.

To find the value of $f(86)$, we need to use the first case again since $86 \geq 1000$. This means that $f(86)=86-3=83$.

Therefore, we can conclude that $f(84)=83$.