CPhill

Given the function f(x)=3x^3+2, find the value of x so that f^{-1}(x)=4

Note that  if    (4, x)    is on the original graph of f(x)  then   (x, 4)  is on the inverse

This implies  that

f(4)  =  x   ....so....

3(4)^3 + 2   =  x  =   194

Here's an example, Nirvana

Suppose  u  is   the vector  (-2,-3)

Suppose v  is the vector  (7,2)

Adding these  vector components we  get the resultant vector , b  =   ( 5, -1)

The absolute value  for  any inputs that are one unit apart  is    25

We can see this by taking the absolute value  of the slope between  any two consecutive  inputs

Example

l  [ -5 - 20 ] / [ -1 - -2]  l  =  l  -25  / 1  l   =    25

I just rounded

The quadratic  evaluates to

[√105 - 1 ]

________    =    sin  θ

     12

So

√3sin  ( arcsin ( [√105 - 1 ]/12 ) )  +  cos (arcsin ( [√105 - 1 ] /12) )  =

√35

 ___   ≈   1.972

   3

3x^2 - 4y^2 + 4x -8y + 4  =  0

3x^2  +y^2  + 4x -3y + 4  = 0

Subtract the first equation from the second and we get that

5y^2  + 5 y   =   0   divide through by  5

y^2 + y  = 0       factor

y ( y + 1)   =   0

Setting   each  factor  to 0   and solving for  y  produces  :

y  =  0      and  y   =  -1

When y  =  0

We have that both equations evaluate to

3x^2 + 4x + 4  = 0    (1)

This has no real solutions for  x   when y  = 0 because the discriminant for (1)  is less than  0

So....there  are no real intersection points  when  y  = 0

When  y  =  -1

The first equation evaluates to

3x^2 - 4 + 4x + 8 + 4  =  0

3x^2 + 4x + 8  = 0       (2)

And the second equation evaluates to

3x^2 + 1 + 4x + 3 + 4  = 0

3x^2 + 4x +  8  =  0    (3)

(2)  and (3)  are the same

Again  when y  =  -1  there are no real solutions for x because  the discriminant is less than 0

So....we  have no  real  solutions   in x  for either y value....

WolframAlpha confirms this   :  https://www.wolframalpha.com/input/?i=3x%5E2+-+4y%5E2+%2B+4x+-8y+%2B+4++%3D+0%2C+++3x%5E2++%2By%5E2++%2B+4x+-3y+%2B+4+%3D+0++

Here's what I get

98, 99 , 100

98  is divisible by 7²

99 is divisible by 3²

100 is divisible by 2²  and 5²

The remaining two  angles must sum to  180 - 72  =  108°

This angle can be  subdivided into   12  equal parts....so  each part =  108 /12   =  9°

So

One angle  =  5(9)  =  45°

And the other  =  7(9) = 63°

So...the smallest angle   = 45°

OK......hope it made sense  ....!!!!

Correct.....good job    !!!

Thanks, guest....one slight correction....

Since  x, y , z   ≥  0

Then the range  is actually  [ 0, infinity  )