CPhill

OK......you  found one answer  (in rads)

To find the other answer......cosine is positive in the 4th quadrant.....so.....the other angle (in rads) will be reference angle of .850 rads in the 4th quad.....so.....

2pi  - .850  ≈ 5.433 rads

Easy to solve these

She  currently has    87 + 77 + 94  =  258 points

She needs  5(90)  =  450 points  for a "90"  average

So....on the next two tests....she needs   450 - 258   =  192  points

So....she needs to average   192 / 2  =   96  on the next two tests

16 + 4x          16 + x

______    =    ______      cross - multiply

  12                  12 - x   

[  16 + 4x ] [ 12 - x ]   = 12 [ 16 + x ]      simplify

192 + 32x - 4x^2  =  192 + 12x      rearrange as

4x^2 - 20x  =  0

4x ( x - 5)   = 0

Only the second factor gives a non-zero   answer

x  = 5

Check

16 + 4(5)           16  + 5

________  =   ________

    12                 12 - 5

36            21

__   =     ___   =   3   .....so.....5 is good  !!!!

12             7

a)    y = x^2 + 3     subtract 3 from both sides         

       y - 3  = x^2     take both roots

     ±√ [ y - 3 ]  =  x           "swap"  x and y

     ±√[ x - 3]  =  y   Note, GM.....this is the inverse.....but  not an inverse "function".....since the original function isn't one-to-one.....then we don't have a true inverse function

b)  y = [ (1/4)x + 6 ] ^3    take the cube root of both sides

 ∛y  = (1/4)x + 6             subtract 6 from both sides

 ∛[y]  - 6   =  (1/4) x         multiply through by 4

4 ∛[y]  - 24   =  x             swap x and y

  4 ∛[x] - 24  = y  =    the inverse

What's the question  ???

x intercept   =   where y = 0  .....so

0  = log₂(x - 3) + 1

-1  =  log₂(x - 3)

In exponential form

2^(-1)  = x - 3

(1/2)  = x - 3

(1/2) + 3  = x intercept  =  3.5

y intercept.....none

Increases from right to left on the domain

Domain  (3, infinity)

Range.....all reals

Asymptotes   x  = 3

End behavior    :  down on the left....up on the right

The inverse  reverses the coordinates  on the original graph

So.....on the original graph   we have the points

 x      y

-1    1/4

0       1

1       4

So...on the inverse.....we have the points

x        y

1/4    - 1

1         0

4         1

Notice that  graph (2)  is the only one possible

I thnk that you might mean this????

tan^-1 ( u)  = θ    means that   tan (θ)  = u  =  u / 1   =  y / x

So....by the Pythagorean Theorem.....

x^2 + y^2  =  r^2

1^2 + u^2  = r^2     take both roots

±√[ 1 + u^2 ]  =  r

So   cos  θ  =    x / r  =   ±1 / √[ 1 + u^2 ]

[ I forgot to take both roots in my first answer ]

tan^-1 (u)   = θ   which implies that

tan (θ)  =  u   =  u / 1

So.....tan =  y / x     which implies that    y = u   and  x  = 1

And   cos ( θ)  =  x / r

So    r  = √ [ x^2 +  y^2 ]  =  √[ 1^2 + u^2 ]  = √ [ 1 + u^2 ]

So....putting all this together

cos ( tan^-1 (u) )  =  cos (θ)   =  x / r  =     1 / √ [ 1 + u^2 ]

THX, EP....