Functions doubt

[eager locust]

for this question I put that angle equal to y
and solving a quadratic I got
-1 <= y <= 1
I get it that y will always be greater than 0
so 0 < y <= 1
I just put cos inverse to get
arccos 0 < arccosy <= arccos 1
to get (pi/2 , 0]
the answer given is [0, pi/2]
I know there's something about arcos being a decreasing function like that.. but I don't get it. can anyone please explain this to me clearly? I'm really bad at understanding functions

[shadow rampart]

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[bold hemlock]

arccos(x) is a decreasing function, so it flips the inequality. As in, if a < x < b and f(x) is a decreasing function, then f(b) < x < f(a).

So, with that, you do get [0, π/2), and their inclusion of π/2 is wrong.

[eager locust]

what if it was an increasing function?

[eager locust]

but why and HOW does a decreasing function "flip" the inequalities? can you please help me visualize this concept?

[scenic prism]

Cengage ahh doubt

[scenic prism]

consider the function $f:[0,1] \to \mathbb{R}$ defined as $f(x) = 1-x$. Would you say the range of $f(x)$ is $[1, 0]$ or $[0, 1]$?

[short holly]

omeganebula

[eager locust]

um sorry i cant figure it out

can u explain

[scenic prism]

its a simple question for you, calculate the range of 1-x over the domain [0, 1]

[eager locust]

if we calculate the end points. at x = 0; fx is 1
and at x=1; fx is 0
should it be [1,0]?

should i flip it?

[scenic prism]

intervals dont work like that unfortunately

[scenic prism]

not without thinking why

so

[eager locust]

the lesser value should be written before the larger one?

[scenic prism]

an interval $(a, b)$ is defined as the set of all real numbers $x$ such that $a < x < b$ correct?

[short holly]

omeganebula

[eager locust]

yes

[scenic prism]

yes, that is the inherent problem

but the way youre approaching inequalities

is also a problem

so

for any function f

can i say if a < x < b, then f(a) < f(x) < f(b)?

[eager locust]

only if fa is less than fb?

[scenic prism]

still not rigorous

want a counterexample?

[eager locust]

yes please

[scenic prism]

take a = -1, x = 0, b = 2, f(x) = x^2

a < x < b but 1 < 0 < 4 is not true

but satisfies your condition of f(a) < f(b)

do you see the problem?

[eager locust]

oh

oh

yeah

then how are we supposed to

[scenic prism]

so you want a stronger condition

[eager locust]

yes

[scenic prism]

any ideas what we could do?

[scenic prism]

to remove these kinds of problems

[eager locust]

like calculating fa fb and fx and comparing them?

[scenic prism]

no no

we are looking for a general result

[scenic prism]

we want to find for what functions this is true

[eager locust]

is this where increasing and decreasing functions come?

[scenic prism]

precisely

[scenic prism]

infact, you may see this as the definition of increasing functions

[eager locust]

thats true for increasing functions and false for decreasing functions am i correct.?.... i wont say ive memorized it but you know i dont get the "feel" of it... can u explain with those examples

[eager locust]

ohh

wait a minute. as a < x < b
for a increasing function f{x}
as value of a increases to x f{a} also increases to f{x}
and as value increases from x to b... f{x} also increases to f{b}

[scenic prism]

i just wanted you to answer based on your intuition

[scenic prism]

yep

now replicate this thinking for decreasing functions

as value of a increases to x, f(a) decreases to f(x)

which implies f(x) < f(a)

also please note here we are talking about strictly increasing and decreasing functions

[eager locust]

okay.. so let me think for decreasing functions. say g{x}
for a < x < b
as value increases from a to x f{a} decreases and becomes f{x}
and as value increases from x to b f{x} decreases to become f{b}
so technically umm f{b} < f{x} < f{a}

i hope im right

[scenic prism]

yep

and if its neither strictly increasing, nor strictly decreasing on the interval (a,b), you cannot comment anything

did you get "feel"?

[eager locust]

hell yeah

[eager locust]

wait what?

a function which is neither increasing nor decreasing...

can you give me an example

[scenic prism]

.

on the interval (-1, 2), the function x^2 is neither strictly increasing, nor strictly decreasing

[eager locust]

oh yeah

so we cant say ANYTHING about such stuff?

[scenic prism]

no, unless you know the explicit behaviour of the function

[eager locust]

i started focusing on these books for maths and phy entirely for my JEE prep... have u completed them? if yes can u help me out? you see i dont have much time left for jee compared to the insane number of questions per chap of cengage.
im trying to choose and focus on them only instead of doing everything
illustrations
concept applications
single correct
numerical type questions

out of these which ones should i focus for maths/physics? illustrations are compulsory so im thinking of skipping concept applications and do either the last 15-20 questions of single correct type OR just numerical type questions? which one to choose

[eager locust]

oh..

[scenic prism]

aw man what did i get myself into

[eager locust]

💀

[scenic prism]

it depends on your level, but I'll say they are not enough for jeea

and the questions are essentially very basic, if your basics are strong enough and you feel the questions get repetitive, move on to harder stuff

if you struggle with the basics, get them cleared first

[eager locust]

yeah the illustrations are too easy for me

but numerical type questions?

[scenic prism]

maybe some questions, but generally not enough

[eager locust]

got it.

but

Iorodov for physics and Black Book for maths

[scenic prism]

i will not comment on them

anyways close the channel, i think we're done here

[eager locust]

alright. thanks again man

+close

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