help? algebra functions

[tight bane]

It says that the solution for a) is 5f(-x)+2f(x)=-2x^3-5x, f(x)=(2/3)x^3+(5/3)x.
b) Verify relation f(-x)=-f(x), any x E real
c) Observe that f(-n) + f(n) = 0, any n E {1,2,...,100}
Calculate S=f(0)=0

The subject has 5f(x)+2f(-x)=2x^3+5x. Why at a), they do 5f(-x)+2f(x)=-2x^3-5x?? how do they get to f(x)=(2/3)x^3+(5/3)x??
At b), i do not understand.
at c) is it Gauss formula??? why don't these solutions show the step by step?

Please if anyone has the patience to help me 😭

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[tight bane]

(if it's easier to visualize the solution) (tag me if answered please)

[wide wren]

for a, note that the given equation means (evaluating at $x\to-x$) $5(x)+2f(-x)=2x^3+5x$, so you get a system of equations in the unknowns $f(x)$ and $f(-x)$

[exotic widget]

Omegabet_

[wide wren]

solving the system for $f(x)$ concludes a

[exotic widget]

Omegabet_

[wide wren]

b just verify the definition of being an odd function

c, by part b, f(-n)=-f(n), hence f(-n)+f(n)=0 for all n, so grouping the terms in this manner shows it's a sum of 0's

[tight bane]

But why does it change - and +? Why does it go from
5f(x)+2f(-x)= 2x^3+5x
to
5f(-x)+2f(x)=-2x^3-5x?

[tight bane]

What was it? i didn't get to that yet. I have to get ready for a comp and a lot of the subjects are things we still haven't been taught

And how do i verify it 😭

[storm solstice]

We have:
5f(x) + 2f(-x) = 2x^3 + 5x
Let's replace x by -x. Then we get:
2f(x) + 5f(-x) = -2x^3 - 5x
You can find f(x) from that system.
For (c), note that f(x) is odd.

[tight bane]

Do we also do f(x)=ax+b, so then 5f(x)=5ax+5b?

Or

[wide wren]

I subbed in -x for x

$5f(-(-x))+2f(-x)=-2(-x)^3-5(-x)\to 5f(x)+2f(-x)=2x^3+5x$

[exotic widget]

Omegabet_

[wide wren]

so you get $\begin{cases}5f(x)+2f(-x)=2x^3+5x\
5f(-x)+2f(x)=-2x^3-5x\end{cases}$

[exotic widget]

Omegabet_

[wide wren]

so just solve the system for $f(x)$

[exotic widget]

Omegabet_

[wide wren]

a function is odd iff $f(-x)=-f(x)$, so just verify both $f(-x)$ and $-f(x)$ are the exact same things