- #1

Albert1

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

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

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In summary, an Integer Function is a mathematical function that maps a set of integers to another set of integers. To find the value of $f(84)$ in an Integer Function, one would plug in the value 84 for the input variable in the function and evaluate the resulting expression. Some common examples of Integer Functions include linear functions, quadratic functions, and exponential functions. Finding the value of $f(84)$ in an Integer Function is important because it can help analyze the behavior and patterns of the function and make predictions about its outputs for other input values. The method for finding the value of $f(84)$ in an Integer Function may vary, but it generally involves plugging in the input value and simplifying the resulting expression using

- #1

Albert1

- 1,221

- 0

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

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

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- #2

I like Serena

Homework Helper

MHB

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Albert said:

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

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

$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} $

Use full induction going down.

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 $f(84) = 997$.

- #3

Albert1

- 1,221

- 0

I like Serena :very smart induction (Clapping)I like Serena said: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 $f(84) = 997$.

- #4

kaliprasad

Gold Member

MHB

- 1,335

- 0

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

- #5

CellCoree

- 1,490

- 0

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 $f(84)=83$.

An Integer Function is a mathematical function that maps a set of integers to another set of integers.

To find the value of $f(84)$ in an Integer Function, you would plug in the value 84 for the input variable in the function and evaluate the resulting expression.

Some common examples of Integer Functions include linear functions, quadratic functions, and exponential functions.

Finding the value of $f(84)$ in an Integer Function can help us analyze the behavior and patterns of the function, and make predictions about its outputs for other input values.

The method for finding the value of $f(84)$ in an Integer Function may vary depending on the specific function, but it generally involves plugging in the input value and simplifying the resulting expression using basic algebraic principles.

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