function composition and inverse.

[mrcl]

[FONT=MathJax_Math]Please help me solve this :
f(x)[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math]x[/FONT]^{[FONT=MathJax_Main]2[/FONT]}[FONT=MathJax_Main]- 10x + 6
[/FONT][FONT=MathJax_Main]
the question is: find the (f_o f ^-1)(x).[/FONT]

 

[lookagain]

[FONT=MathJax_Math]Please help me solve this :
f(x)[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math]x[/FONT]^{[FONT=MathJax_Main]2[/FONT]}[FONT=MathJax_Main]- 10x + 6
[/FONT][FONT=MathJax_Main]
the question is: find the (f_o f ^-1)(x).[/FONT]

\(\displaystyle f(x) = x^2 - 10x + 6 \ \ \) has no inverse, because it is not a one-to-one function. No parabolas are one-to-one. Therefore, all parabolas have no inverses.

 

[Deleted member 4993]

Just like sin(x) or cos(x) has corresponding inverse functions - with restricted domain and range - parabolas and other non-monotonic functions have corresponding inverse functions with restricted domain and ranges.

 

[HallsofIvy]

Since the graph of this is a parabola, with one increasing and one decreasing (and so one-to-one) piece, you could separate it into to functions at the vertex, giving an inverse function for each. That is NOT the same as " (f_o f ^-1)(x)" but is the best you can do.