I'm unable to understand why option c is correct.Even though option a seems correct If A and B are the points of intersection of $y=f(x)$ and $y=f^{-1}(x)$ then
(a) A and B necessarily lie on the line $y=x$
(b) A and B must be coincident
(C)Slope of line AB may be $-1$
I approached by  considering two point
$P(x,y)$ and $Q(y,x)$ on $y=f(x)$ and $y=f^{-1}(x)$.
If these two  functions curves intersect at point $A$ then $P$ and $Q$ must coincides thus Point $A$ must be of form $(a,a)$,hence lies in line $y=x$.
 A: It's a confusingly worded problem because the graphs of $f$ and $f^{-1}$ don't necessarily have exactly two intersections, so saying "$A$ and $B$ are the points of intersection" is not really meaningful.
For example $x\mapsto x+1$ has no intersection with its inverse function.
$x\mapsto 2x$ has one point of intersection with its inverse function -- namely $(0,0)$.
$x\mapsto -x$ equals its inverse, so it "intersects" it at every point of its graph. But there are also cases with discrete (and therefore in some sense more "honest") intersections:
$x\mapsto -x^3$ has exactly three points of intersection with its inverse: $(-1,1)$, $(0,0)$, and $(1,-1)$. Here in particular you can see that the intersections don't have to lie on the line $x=y$, and that option (C) can be true.
$x\mapsto \sin(x)+x$ has infinitely many intersections with its inverse. They all lie on the line $x=y$, but if we take $x\mapsto \sin(x)-x$ instead the intersections rather lie on the line $y=-x$.
Finally, you can take the curve $y=\frac{\sin(x)+\sqrt{x^2+1}}2$ and rotate it by 45° around the origin. You get the graph of a function $\mathbb R\to\mathbb R$ whose countable infinity of intersections with its inverse function don't even lie on a common line.
A: Consider the function $f(x)=-x$. This function tells us that (a) and (b) are false, whereas (c) is true.
A: C is correct. Notice the question is about two curves that intersect in at least two points. Two curves which intersect at only one point or no points is outside the scope of the problem. Now go to your favorite online plotting calculator (I used https://www.desmos.com/calculator), and plot the two curves, y=(x-4)^2, and y=sqrt(x)+4 (and y=-sqrt(x)+4 if you wish). Whether or not you plot the top, bottom, or both of the inverse curves you can connect intersecting points to make lines which are not at 45 degrees.
Now of all the possible equations and their inverse that you may plot, some of these intersecting curves will make for a line with a slope of -1. This is a sneeky question because of the word, "necessarily" in 'A', and the word "must" in 'B'. 'C' is the correct answer because it uses the words, "may be".
