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It is given that the function

$f(x) = \sqrt{(mx + 7)} - 4$

$x \geq -\frac{7}{m}$

and its inverse do not intersect, and that neither intersect the line $y = x$.

How could one determine the set of possible values for the positive constant m?

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When you talk of $y$, presumably you mean $f(x)$. Since $f$ is continuous, to avoid having you must either have $f(x)\gt x$ for all $x$ or have $f(x) \lt x$ for all $x$. This explicitly keeps $f(x)$ from meeting $y=x$. You can set $f(x)=x$ and find what values of $m$ make it impossible to satisfy. The condition that $f(x) \ne f^{-1}(x)$ is covered as you must have $f(x)\gt x \gt f^{-1}(x)$ or the reverse.

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Yes, y is specified as being equal to f(x). My teachers over the years have generally used 'y' and 'f(x)' interchangeably. –  tzxn3 May 25 '12 at 11:47
    
The only appearance of $y$ in the post is in the line $y=x$*, which looks correct to me. (On the other hand, to say that *$f$ and $f^{-1}$ meet to mean that the graphs of the functions $f$ and $f^{-1}$ intersect, is questionable. Likewise for the graph of $f$ and the line $y=x$.) –  Did May 25 '12 at 12:14
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