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?
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?
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.