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1. Some functions that aren't invertible can be made invertible by restricting their domains. For example, the function \(x^2\) is invertible if we restrict \(x\) to the interval \([0,\infty)\), or to any subset of that interval. In that case, the inverse function is \(\sqrt x\). (We could also restrict \(x^2\) to the domain \((-\infty,0]\), in which case the inverse function would be \(-\sqrt{x}\).)

Similarly, by restricting the domain of the function \(f(x) = 2x^2-4x-5\) to an interval, we can make it invertible. What is the largest such interval that includes the point \(x=0\)?

 

2. The function \(f(x) = \frac{cx}{2x+3}\)
satisfies \(f(f(x))=x\), \(x\ne -\frac 32\) for all real numbers . Find \(c\).

Guest Jun 20, 2017
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1.  The vertex of this parabola is (1, -7)....so.....restricting the domain to  (-infinity, 1 ] will  make the function invertible...... and  x  = 0  lies within this interval

 

2. If f ( f(x) )  = x        we can write

 

c ( (cx) / [2x + 3] )

________________        =  x   multiply  throgh by the denominator on the left side

2 ( (cx/ [2x + 3]) + 3

 

 

c( (cx)) / [2x + 3] )  = x  [ 2 ( (cx) / [2x +  3] ) + 3]     simplify

 

c^2x / [2x + 3]  =  x [ 2cx + 6x + 9 ] / [2x + 3]

 

c^2x  =  2cx^2 + 6x^2  + 9x

 

2cx^2 + 6x^2 + (9 - c^2) x  = 0

 

Note....that for any x, this will equal 0 whenever

 

2c +  6 + (9-c^2)  = 0    

 

-c^2  + 2c + 15  = 0     multiply through by -1

 

c^2  -  2c  - 15  = 0  factor

 

(c - 5) (c + 3)  = 0

 

So....setting each factor to 0 and  solving for c we have the possible values c = 5  or c = -3

 

Test c = 5  in  f(f(x))

 

5[5x /[2x + 3] ]                         [  25x ] / [2x + 3]

_______________     =        _________________  =   25x  /  [ 16x + 9 ]    

2 [ 5x / [2x + 3]] + 3             [10x + 6x + 9]/ [2x + 3]

 

So  c = 5  is not a solution

 

Test  c  = -3  in  f (f (x))

 

-3 [ -3x / [2x + 3] ]                  [ 9x] / [2x + 3]                        9x

________________     =     __________________   =      ___     =    x    

2 [ -3x / [2x + 3] ] + 3           [-6x + 6x + 9] / [2x + 3]             9

 

So....  c  =  -3  

 

 

 

cool cool cool

CPhill  Jun 20, 2017

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