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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 $y=x$.

LogLogLogLogx
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MUZAMIL ANWAR
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    What happens if $f$ is its own inverse? – Giorgos Giapitzakis Jul 03 '21 at 11:45
  • Continuing Giorgos' suggestion, for functions that are their own inverse, you can think of $y = -x$ or $y = \frac{1}{x}$. Have a look at how they behave with respect to the statements (a), (b), and (c). – Mankind Jul 03 '21 at 12:48

3 Answers3

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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.

Troposphere
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  • The graphs for a general function don't necessarily have two points of intersection; but since the question assumes that does, isn't that implicitly defining as such a function? – gidds Jul 03 '21 at 20:02
  • @gidds: That's an interesting proposal -- but it seems unlikely that it's the intention of the problem if option C is stated to be the correct answer. (In that case the function would have to be non-continuous). – Troposphere Jul 03 '21 at 20:09
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Consider the function $f(x)=-x$. This function tells us that (a) and (b) are false, whereas (c) is true.

Joe
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  • Hi Joe, I feel this would be a naive thing to do. An example maybe used to disprove wrong statements, but it is not sufficient to say with certainty that the true statements are true. This is due to the fact that in the example , there maybe more 'nicer' properties which causes 'c' to be true as well (while it may not be generally) – tryst with freedom Jul 03 '21 at 20:51
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    @Buraian: But the function $f(x)=-x$ is a counterexample to "(a) $A$ and $B$ necessarily lie on the line $y=x$", and likewise for (b). These are universal statements—and so we just need one counterexample to disprove them. Also, (c) merely asserts the existence of a function such that the slope of AB is $-1$, and $f(x)=-x$ is such a function. – Joe Jul 03 '21 at 21:43
  • Hi Joe, if you see the first part which I wrote, you will find that I am agreeing with you there. I didn't read the 'c' part carefully, I missed the 'may be' part that you have mentioned now. Now I see what you said is correct. – tryst with freedom Jul 03 '21 at 21:46
  • @Buraian: Okay, thanks. – Joe Jul 03 '21 at 21:47
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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".

Scot Parker
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