# Finding $f(84)$ with the Defined Function $f$

• MHB
• anemone
In summary, the function $f$ is defined on the set of integers and satisfies the first condition of $n \geq 1000$ and the second condition of $n < 1000$. To find $f(84)$, we use the first condition and substitute $84$ for $n$, giving us $84-3=81$. Therefore, $f(84)=81$.
anemone
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MHB
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The function $f$ is defined on the set of integers and satisfies

$f(n)=\begin{cases} n-3, & \text{if} \,\,n\geq 1000 \\ f(f(n+5)), & \text{if}\,\, n< 1000 \end{cases}$

Find $f(84)$.

Last edited:
anemone said:
The function $f$ is defined on the set of integers and satisfies

$f(n)=\begin{cases} n-3, & \text{if} \,\,n\leq 1000 \\ f(f(n+5)), & \text{if}\,\, n< 1000 \end{cases}$

Find $f(84)$.

I think 1st condition should be $>=$

Ah, thanks to kaliprasad for pointing it out! Sorry all! I made a blunder, I think for those who know me well, you can tell this wasn't the first time I made a typo in my posting, hehehe...but, I apologize. This shouldn't happen in the first place, I should have checked both before and after posting too!

anemone said:
Ah, thanks to kaliprasad for pointing it out! Sorry all! I made a blunder, I think for those who know me well, you can tell this wasn't the first time I made a typo in my posting, hehehe...but, I apologize. This shouldn't happen in the first place, I should have checked both before and after posting too!

No one is perfect, and only those who do things put themselves at the risk of making a mistake. When I see that someone contributing to a site has possibly made a typo or other error in a post, I tend to send them a PM so they can correct it with minimal embarrassment. :)

As for the value of f(n) for n less than 1000 can be computed if we now for n+5 so let us compute f(n) for n = 1004 to 995 downwards
$f(1004) = 1004- 3 = 1001$
$f(1003) = 1003 - 3 = 1000$
$f(1002) = 1002-3 = 999$
$f(1001) = 1001 - 3 = 998$
$f(1000) = 1000-3 = 997$
$f(999) = f(f(1004)) = f(1001) = 998$
$f(998) = f(f(1003)) = f(1000) = 997$
$f(997) = f(f(1002)) = f(999) = 998$
$f(996) = f(f(1001)) = f(998) = 997$
$f(995) = f(f(1000)) = f(997) = 998$

Now as 84 + 915 = 999 or 84 + 183 * 5 = 999 So let us evaluate f(999-5k) for some k

we have $f(994) = f(f(999)) = f(998) = 997$
$f(989) = f(f(994)) = f(997) = 998$
$f(984) = f(f(989)) = f(998) = 997$
from the above we see that $f(999-kn)$ is $998$ if k is even and is $997$ if k is odd

as $999- 84 = 183 * 5$ so we get $f(84) = 997$

## 1. What is the purpose of finding f(84) with the defined function f?

The purpose of finding f(84) with the defined function f is to evaluate the output of the function at a specific input value of 84. This can help us understand the behavior of the function and its relationship between input and output values.

## 2. How do we find f(84) with the defined function f?

To find f(84) with the defined function f, we simply plug in the input value of 84 into the function and solve for the output value. This can be done by substituting 84 for the variable in the function and using algebraic techniques to simplify the expression.

## 3. What is the importance of finding f(84) with the defined function f?

Finding f(84) with the defined function f is important because it allows us to make predictions and analyze the behavior of the function at a specific input value. This information can be useful in various fields such as mathematics, physics, and economics.

## 4. Can we find f(84) with the defined function f for any value of 84?

Yes, we can find f(84) with the defined function f for any value of 84 as long as the function is defined for that input value. If the function is not defined for 84, then we cannot find the output value.

## 5. How does finding f(84) with the defined function f differ from finding f(x) with the same function?

Finding f(84) with the defined function f is similar to finding f(x) with the same function, except that we are evaluating the function at a specific input value of 84 instead of a general input value of x. This allows us to get a specific output value instead of a variable expression.

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