Inverse of a Function

Transcript

0:00Introduction

Hello everyone! Okay, one small puzzle before we start. The graph of f inverse is the mirror image of f in the line y equals x. So where can a function meet its own inverse? Only on that mirror line, right? Now take f of x equals minus x cubed. It meets its inverse at three points, and two of them are not on the line y equals x! So how is that possible? Today: the inverse of a function, how to find it, its graph, and where this mirror idea breaks. Look, inverse questions come in JEE Main again and again. So, let's begin.

0:41The inverse undoes f

So, take a function f from A to B, okay? And let it be a bijection: one-one and onto, both together. Now, f takes an input x to an output y. The inverse goes back, from B to A, okay? We write f with a small minus one on top, and read it, f inverse. The rule: f inverse of y equals x, if and only if f of x equals y; that is the double arrow. Clear? Now think. Start with x, apply f, and then apply f inverse. Where do you land? Back at x, yes! So f inverse after f is I A, the identity on A, okay? The small circle means composition. And f after f inverse is I B, the identity on B, okay? So the inverse simply undoes f. That's it.

1:30When does f⁻¹ exist?

Alright, but can every function be reversed? Suppose two different inputs give the same output y. Then what is f inverse of y? Well, it would need two answers at once, and a function cannot do that, right? So f must be one-one. And if some y in the set B is never reached, f inverse of y has nowhere to go, so f must be onto too. So f inverse exists exactly when f is one-one and onto. No bijection, no inverse! And when the inverse exists, there is only one, and it is itself a bijection, from B back to A, okay? And the domain and range simply swap. The domain of f inverse, written Dom on screen, is the range of f. And the range of f inverse is the domain of f. Really simple, got it?

2:20Two rules, and a trap

Two quick rules. First, the inverse of f inverse is f itself. Undo the undo, and you are back to f, okay? Second, g circle f: first f, then g. So its inverse is g inverse circle f inverse, in the same order, right? No, no, careful! This is the JEE trap. Think about undoing. f was done first, and g last. So to undo, undo g first: g inverse first, then f inverse. Composition is written right to left, so the inverse of g circle f is f inverse circle g inverse. The order reverses! With me?

2:59The recipe for f⁻¹

Now, how do we actually find f inverse? Just a simple recipe. Write y equals f of x, and solve this equation for x, in terms of y. That gives the input x for the output y, which is exactly f inverse of y. Then rename y as x, and that is f inverse of x, okay? Simple, right? Just write, solve, rename. We will use this recipe in example one.

3:27The graph: a mirror image in y=x

Now the graph. If the point a comma b is on f, then f of a is b. So f inverse of b is a, and the point b comma a is on f inverse. The coordinates simply swap, okay? Look at the figure. The blue curve is e to the power x; the orange one is its inverse, l n x, the natural log. The blue dot is one comma e. So which point must be on the orange curve? The point e comma one, yes! The orange dot. And swapping x and y is a reflection in the line y equals x, the dashed line. So the graph of f inverse is the mirror image of the graph of f in that line. Exactly!

4:11Topper's trick: where f meets f⁻¹

Now a trick that toppers use. Suppose you must solve f of x equals f inverse of x. Hard, right? First find f inverse, then solve a messy equation. But look: if f is strictly increasing, f and f inverse meet only on the line y equals x. The arrow on screen means, implies. So simply solve f of x equals x, instead of the harder equation. You never need the formula for f inverse at all. Clear?

4:41Careful: decreasing functions

Now, back to our puzzle. f of x equals minus x cubed, and its inverse is minus the cube root of x. But this f is decreasing, not increasing, okay? So does our trick still work here? Okay, let's test it. Set them equal: minus x cubed equals minus the cube root of x. The minus signs cancel; now cube both sides: x to the power nine equals x. So x is zero, or x to the eight is one, giving plus or minus one. The plus-minus sign means both, okay? So the two graphs meet at three points: zero comma zero, one comma minus one, and minus one comma one. Look at the green dots. Two of them are away from the dashed line y equals x! So the answer to my question is no; the trick fails here. Puzzle solved. Blindly solving f of x equals x gives only x equals zero, and you lose two answers. So the trick is only for increasing functions. Got it?

5:42Example 1: inverse of f(x) = 3x - 5

Right, on to example one. Find the inverse of f from R to R, with f of x equals three x minus five. The double-lined R means the real numbers. Pause the video, and try it with the recipe. Done? Okay, let's check. First, y equals three x minus five. Now solve for x. Add five to both sides, and divide by three. So x equals y plus five, over three. Last step, rename y as x. So f inverse of x equals x plus five, over three. Done! And remember, toppers always check. Put f inverse inside f. Three times x plus five over three, minus five. The threes cancel, giving x plus five minus five, just x. We got x back, so the inverse really undoes f. Clear?

6:39Example 2: where x squared - 2x + 2 meets its inverse

Example two, JEE level. f goes from one to infinity, into one to infinity, with f of x equals x squared minus two x plus two. The square bracket includes one; the sideways eight is infinity. Solve f of x equals f inverse of x. Pause and try. Hint: the topper's trick. Done? You know, most students first find f inverse, then solve a messy equation. Long, and easy to get wrong. Is there a shorter way? Yes, check whether f is increasing! Complete the square: x minus one, whole squared, plus one. As x grows from one onwards, this square grows too. So f is strictly increasing here, okay? Increasing, so f and f inverse meet only on the line y equals x. So we just solve f of x equals x. x squared minus two x plus two equals x. Bring the x to the left: x squared minus three x plus two equals zero. This factors as x minus one, times x minus two. So x equals one and x equals two, both in the domain. Quick check: f of one is one, and f of two is two. Absolutely correct! And we never found f inverse! But remember minus x cubed: this works only because f is increasing.

8:08Quick revision: when it exists, the recipe, mirror in y = x

So, quick revision. An inverse exists only for a bijection, and it undoes f. The recipe: write y equals f of x, solve for x, and rename y as x. And the graph is a mirror image in the line y equals x; an increasing f meets its inverse only there. Now go and try the practice problems on this page. All the best!