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# I have 2 questions

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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 $$c$$.

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

Jun 20, 2017