hectictar

In this context  g^-1(x)  does not mean  g(x)  raised to the power of negative 1 .

It means the inverse function of  g(x) . We can find  g^-1(x)  like this....

g(x)   =   9^x  -  3

                                        Instead of  g(x) ,  write  y .

y  =  9^x  -  3

                                        Now we want to solve this equation for  x  .

y + 3   =   9^x

log(y + 3)   =   log(9^x)

log(y + 3)   =   x log 9

log(y + 3) / log 9   =   x

                                        To get the inverse function, swap  x  and  y .

log(x + 3) / log 9   =   y    This is the inverse function. We can say...

g^-1(x)   =   log(x + 3) / log 9

So...    g^-1(x)  does not mean  1 / g(x) .

Neither does  sin^-1x  =  (sin x)^-1   ,  even though  sin²x  =  (sin x)²

This is something that has bothered me before. Maybe there is a reason why it is okay and I just don't understand it......?  

a.

distance between warehouse and Abby   =   distance between warehouse and Cardenas

Let  (x, y)  be the coordinates of the warehouse.

distance between  (x, y)  and  (-3, -2)   =   distance between  (x, y)  and  (3, 6)

√[  (x + 3)² + (y + 2)² ]   =   √[  (x - 3)² + (y - 6)²  ]

(x + 3)² + (y + 2)²   =   (x - 3)² + (y - 6)²

x² + 6x + 9  +  y² + 4y + 4   =   x² - 6x + 9  +  y² - 12y + 36

6x + 4y + 4   =   -6x - 12y + 36

16y   =   -12x + 32

y  =  -3/4x + 2         The warehouse can be any point along this line.

temperature   =   f(x)   =   3.8 cos( πx / 20 ) + 2.2

The lowest temperature will occur when  cos( πx / 20 )  is at its lowest value, which is   -1  .

3.8(-1) + 2.2   =   -3.8 + 2.2   =   -1.6

The lowest temperature is  -1.6  degrees Celsius.

(a)

Remember, similar triangles can have different side lengths.

In order for two triangles to be similar, the angles must be the same.

Since  ∠DCE  and  ∠BCA  are vertical angles, they have the same measure.

Since  ∠CDE  and  ∠CBA  are right angles, they have the same measure.

And since two of the angles are the same, the third angle must be the same,

so  △ABC is similar to △EDC  by the Angle-Angle similarity rule.

(b)

To solve this, we have to know that △ABC is similar to △EDC .

Because △ABC is similar to △EDC , we can say that

22/100  =  32/x   , where  x  is the length of  AB  in feet.

x  =  3200/22   ≈   145.45

So the width of the river is about  145.45  feet, just as you found!

Remember, the Pythagorean theorem says...

(leg)² + (other leg)²   =   (hypotenuse)²

Since the legs of this triangle are sides of a square, they must be the same length.

Let's call the length of each leg  s , and plug in  5√2  for the hypotenuse.

s² + s²   =   ( 5√2 )²

                                         Now we can solve this equation for  s .

2s²   =   ( 5√2 )²

2s²   =   5√2 * 5√2

                                         Multiplication can be done in any order.

2s²   =   5 * 5 * √2 * √2

                                         And  √2 * √2  = 2

2s²   =   25 * 2

                                         Divide both sides of the equation by  2 .

s²   =   25

                                         Take the positive, since  s  is side length, square root of both sides.

s   =   5

And   125x^6   =   125 * x^6   =   125 * x * x * x * x * x * x  

the maximum value occurs when...

t   =   -8 / [ 2(-1) ]

t   =   -8 / -2

t   =   4

When  t  =  4 ...

-t² + 8t - 4

=  -4² + 8(4) - 4

=  -16 + 32 - 4

=  12

original price  -  35% of original price   =   $247

                                           Let the original price be  p  dollars.

p  -  35%  of  p   =   247

p  -  0.35 * p   =   247

                                           Factor  p  out of the terms on the left side.

p( 1 - 0.35 )   =   247

p( 0.65 )   =   247

                                           Divide both sides of the equation by  0.65 .

p   =   247 / 0.65

p   =   380

t   =   - (-9) / [ 2(1) ]

t   =   9/2

t   =   4.5

x² + bx + 1/3

=   x² + bx + (b/2)² - (b/2)² + 1/3

=  (x + b/2)² - (b/2)² + 1/3

Now we can see that...

-(b/2)² + 1/3   =   1/12

-(b/2)²   =   1/12  - 1/3

-(b/2)²   =   -1/4

(b/2)²   =   1/4

b²/4   =   1/4

b²   =   1

b   =   ± 1

b  is a negative number, so   b = -1