A level maths help

[zenith ridge]

How would i find the range of part b . ive done part a and got 3(x-2)^2+8 but idk where to go from there.

[buoyant rune]

so just to like point out

obviously the quadratic in a is the denominator in b so g(x) you can write as

1/(3(x-2)^2 + 8)

right

[zenith ridge]

yep

[buoyant rune]

now that (x-2)^2

what can you tell me about the range of that

what can you tell me about the output of it

[zenith ridge]

that it has to be positive

[buoyant rune]

yep or 0

[zenith ridge]

yeah

if x=2

[buoyant rune]

now 3 times that obviously wont make a difference

[zenith ridge]

yeah

oh so the lowest is 1/8

[buoyant rune]

exactly

and it can be 1/(8 + positive)

[zenith ridge]

so its g(x) > or euqla to 1/8?

equal

[buoyant rune]

ah not quite so

[zenith ridge]

ok

[buoyant rune]

lowest actually isnt right i was wrong

[zenith ridge]

thats fine

[buoyant rune]

so if you made the 3(x-2)^2 positive

say instead it was like 3

your g(x) would be 1/11 right

so smaller than your 1/8

[zenith ridge]

say instead what was 3?

[buoyant rune]

your 3(x-2)^2

[zenith ridge]

oh ok yhyh

[buoyant rune]

just pretend you put in a value so it was 3

[zenith ridge]

ok

[buoyant rune]

then your results gonna be 1/11

[zenith ridge]

yeah

sa

[buoyant rune]

smaller than your 1/8

[zenith ridge]

yeap

yep

[buoyant rune]

what if you plugged in a larger number

like as you make 3(x-2)^2 bigger and bigger

whats gonna happen to your 1/(3(x-2)^2 + 8)

[zenith ridge]

that itll just get smaller and smaller

[buoyant rune]

yep and importantly

itll 'approach' a number

[zenith ridge]

0?

[buoyant rune]

like it wont go off to - infinity

perfect

[zenith ridge]

ah ok

[buoyant rune]

itll approach 0

[zenith ridge]

so is the lim thing?

[buoyant rune]

so g(x) is at most 1/8

[zenith ridge]

ok

[buoyant rune]

but also greater than 0

[zenith ridge]

ah

ok i see

so its greater than zero but less than or equal to 1/8?

[buoyant rune]

perfect

[zenith ridge]

ok cool that makes sense now

thanks a lot

[buoyant rune]

no worries!