how do I do this (1 Viewer)

Sylfiphy

New Member
not to be dumb or anything but can someone explain how to work this out

If, f(x)= f(x+1) + x^2 , and f(3)=7, evaluate f(1)

scaryshark09

Well-Known Member
first let x = 2

f(2) = f(2+1) + 2^2
f(2) = f(3) + 4
f(2) = 7 + 4 [ since f(3)=7 ]
f(2) = 11

now let x = 1

f(1) = f(1+1) +1^2
f(1) = f(2) + 1
f(1) = 11 + 1 [ since f(2)=11 ]
f(1) = 12

Sylfiphy

New Member
first let x = 2

f(2) = f(2+1) + 2^2
f(2) = f(3) + 4
f(2) = 7 + 4 [ since f(3)=7 ]
f(2) = 11

now let x = 1

f(1) = f(1+1) +1^2
f(1) = f(2) + 1
f(1) = 11 + 1 [ since f(2)=11 ]
f(1) = 12
yeah but maybe I’m js tweakin here but the answer sheet says the answer is -6 ??? how did they get that cuz I did the same working as u

A Morgan Freeman

Member
not to be dumb or anything but can someone explain how to work this out

If, f(x)= f(x+1) + x^2 , and f(3)=7, evaluate f(1)
Alright
First, you rearrange it into f(x + 1) = f(x) - x^2, I swapped the order cause it is easier to work with

f(3) = f(2 + 1) = 7
Now from the f(x+1) equation you can say f(2 + 1) = f(2) - 2^2 = 7
f(2) - 4 = 7

so f(2) = 11
f(2) = f(1 + 1) = 11

From the f(x + 1) function again, f(1 + 1) = f(1) -1^2 = 11
f(1) - 1 = 11

So f(1) = 12

If there are any errors here I am sorry, I am kinda sleepy, but the logic should lead to the right answer.
I haven't seen a question like this in ages.

A Morgan Freeman

Member
yeah but maybe I’m js tweakin here but the answer sheet says the answer is -6 ??? how did they get that cuz I did the same working as u
oh bruh i didn't see this lol

Sylfiphy

New Member
oh bruh i didn't see this lol
Its chill lmfao but I swear this q is making me so mad rn cuz I know it’s easy

A Morgan Freeman

Member
Its chill lmfao but I swear this q is making me so mad rn cuz I know it’s easy
Dude just continue practicing these goofy questions and you will ace them in no time

scaryshark09

Well-Known Member
yeah but maybe I’m js tweakin here but the answer sheet says the answer is -6 ??? how did they get that cuz I did the same working as u
Yeah that’s definitely wrong lol

Luukas.2

Well-Known Member
You can prove the answer is wrong by working from it...

\bg_white \begin{align*} \text{Put f(1) = -6 into} \qquad f(x) &= f(x+1) + x^2 \qquad \text{taking x = 1} \\ f(1) &= f(2) + 1^2 \\ -6 &= f(2) + 1 \\ f(2) &= -7 \\ \\ \text{Put f(2) = -7 into} \qquad f(x) &= f(x+1) + x^2 \qquad \text{taking x = 2} \\ f(2) &= f(3) + 2^2 \\ -7 &= f(3) + 4 \\ f(3) &= -11 \qquad \implies \qquad \text{f(1) = -6 is inconsistent with the given statement that f(3) = 7} \end{align*}

A Morgan Freeman

Member
You can prove the answer is wrong by working from it...

\bg_white \begin{align*} \text{Put f(1) = -6 into} \qquad f(x) &= f(x+1) + x^2 \qquad \text{taking x = 1} \\ f(1) &= f(2) + 1^2 \\ -6 &= f(2) + 1 \\ f(2) &= -7 \\ \\ \text{Put f(2) = -7 into} \qquad f(x) &= f(x+1) + x^2 \qquad \text{taking x = 2} \\ f(2) &= f(3) + 2^2 \\ -7 &= f(3) + 4 \\ f(3) &= -11 \qquad \implies \qquad \text{f(1) = -6 is inconsistent with the given statement that f(3) = 7} \end{align*}
I am starting to think that f(1) = -6 might be wrong

rev668

New Member
not to be dumb or anything but can someone explain how to work this out

If, f(x)= f(x+1) + x^2 , and f(3)=7, evaluate f(1)
To evaluate $$f(1)$$ using the given functional equation $$f(x) = f(x+1) + x^2$$ and the initial condition $$f(3) = 7$$, we can employ a method that involves iteratively substituting values and recursively solving the equation.

Let's start by utilizing the initial condition to find $$f(2)$$:
$f(2) = f(3) + 2^2 = 7 + 4 = 11$

Now, having found $$f(2)$$, we can proceed to find $$f(1)$$:
$f(1) = f(2) + 1^2 = 11 + 1 = 12$

This approach leverages the recursive nature of the given functional equation, where each $$f(x)$$ is dependent on $$f(x+1)$$ and the value of $$x^2$$. By successively substituting values, we can trace back to find $$f(1)$$ from the initial condition.

It's worth noting that this process can be extended further to evaluate $$f(0)$$, $$f(-1)$$, and so forth, depending on the domain of the function and the range of values of interest.

Therefore, the value of $$f(1)$$ is $$12$$. This solution method provides a systematic approach to solving for $$f(1)$$ and showcases the step-by-step process involved in leveraging the given functional equation and initial condition to find the desired value.

ExtremelyBoredUser

Bored Uni Student
yeah but maybe I’m js tweakin here but the answer sheet says the answer is -6 ??? how did they get that cuz I did the same working as u
dw bro some answer sheets esp random ones off online be dodgy as hell, my stupid ass spent an hour on a question rigorously proving i was right and assumed i was wrong, then just realised the answers were wrong. Best thing u can do is just to ask other ppl and if ur acc wrong, theyll most likely point it out.