CPhill

Let . Find all real numbers  such that . 

So...we want to find this....

x² - 2x  = (x² - 2x)² - 2(x² - 2x)         expand

x² - 2x = x⁴ - 4x³ + 4x² - 2x² + 4x     simplify

x² - 2x = x⁴ - 4x³ + 2x²  + 4x            subtract x² - 2x from both sides

0 = x⁴ - 4x³ + x²  + 6x                      rearrange and factor

x (x³ - 4x² + x + 6) = 0

One solution is x = 0

The others are shown in this graph......https://www.desmos.com/calculator/pfvqpci5oa

And the other solutions are x = -1   , x = 2 and x = 3

So, the solurions in order are

x = -1, 0, 2 and 3

 

Let  and . Find  if  for all .

(f o g)(x)   says to take the function "g" and put it into "f"...so we have

2[3x + c ] + 7 =

6x + 2c + 7

(g o f)(x)   says....wait for it......to take "f" and put it into "g"....so we have

3[2x + 7] + c  =    6x + 21 + c

And we want to set these equal and solve for c

6x + 2c + 7 = 6x + 21 + c         subtract 6x from both sides

2c + 7  = 21 + c                        subtract c, 7 from both sides

c = 14

 

Suppose  and . Find 

b(3) = 2 - (3)³  =   2 - 27   = -25

So.... a(b(3) = a(-25) =  3(-25) - 7 = -75 - 7  = -82

And a(3) = 3(3) - 7 = 9 - 7  = 2

So b(a(3)) = b(2) = 2 - (2)³  = 2 - 8  = -6

Therefore...... a(b(3) - b(a(3)) =   -82 - (-6)  =  -82 + 6  =  -76

 

. If , then what is ?

So...we're substituting b for a and 3 for b in the function...so we have

2b - 3(3)² + 7 = 90

2b - 27 + 7 = 90

2b - 20 = 90      add 20 to both sides

2b = 110            divide both sides by 2

b = 55

 

Here's the graph.......https://www.desmos.com/calculator/juj2ievcfv

This graph has a period of about 5.73 degrees....it attains a max of 1.97 at .314 degrees and every ±5.73 degrees on either side of this.