Mini-Lecture on Inverse Notation

by Mr. Hansen, Feb. 2010

 

Today we are going to learn about “inverse notation.” Your book does not use this terminology, perhaps because the authors felt that freshmen couldn’t handle it, but since you need to know the inverse notation for Algebra II and Precalculus, you might as well start learning it now.

 

We will begin with a silly example. Imagine that there is a basketball team, the Denver Dirtballs, with 5 starting players whose jersey numbers are as follows:

 

            18        Dale Dribbler

            27        Bill Ballhaug

            32        Fred Fumbler

            36        Steve (a.k.a. Slim) Shooter

            41        Pete Passer

 

Since these are silly names, the sportswriter (Frannie Fann) observing the game makes all her notes using jersey numbers: “32 p. to 36, assist, 3 pts.; 41 steals inb. pass, p. to 27, p. to 32, turnover.” You get the idea. The point is, when Frannie writes up her notes later in the evening, she has to use a function to translate all the jersey numbers back into names. The finished paragraph would look something like this:

 

“At a particularly dramatic moment in the third quarter, Fred Fumbler passed to Slim, who nailed the 3-pointer. On the in-bound pass, Pete Passer immediately stole the ball, passed to Ballhaug, who foolishly relinquished the ball to Fumbler. This time, Fumbler was true to form and lost the ball. The rest of the evening was a disaster as the Dirtballs lost yet again.”

 

OK, do you see what’s happening here? Frannie has a function table that allows her to translate jersey numbers into player names. Mathematically, we would denote this as follows:

 

PlayerName(18) =“Dale Dribbler”

PlayerName(27) = “Bill Ballhaug”

PlayerName(32) = “Fred Fumbler”

PlayerName(36) = “Steve (a.k.a. Slim) Shooter”

PlayerName(41) = “Pete Passer”

 

We will read these as follows: “Player name of number 18 equals Dale Dribbler. Player name of number 27 equals Bill Ballhaug. Player name of number 32 equals Fred Fumbler.” Do you get the idea?

 

(And, if you are alert, you might see the parallel with sines, cosines, and tangents. When we write  we will read that as, “Cosine of 32 degrees equals approximately .848.” The cosine function is a function in the same way that the PlayerName function is a function.)

 

So far, so good. However, what happens if Frannie goes to the locker room for a post-game interview, and on the way a fan from the opposing team asks her what Slim’s jersey number is? As a sportswriter, she should surely know that, right? However, she cannot use the PlayerName function, since the PlayerName function takes a jersey number as input and returns a player name as output. Here, Frannie wants to perform the inverse operation: given a player name, return the jersey number.

 

With 5 names, it is not too challenging to run the function in either direction. In fact, basketball is fairly easy to follow even when there are more players than the 5 starters to keep track of. However, perhaps you see where we are going with this idea. Imagine, for example, a football team with dozens of players who rotate on and off the field over the course of the game. Do you see how having a quick lookup function for finding player names would be essential, as well as a quick inverse lookup function for finding jersey numbers for particular players? Coaches learn to keep these functions in their heads at all times, but spectators and sportswriters usually need a printed program to help them.

 

The notation we use for “inverse function” is a little odd-looking. We write a superscript −1 next to the function name to indicate that we are operating in reverse (i.e., in the inverse direction).

 

What we write: PlayerName−1(“Bill Baulhaug”) = 27.

How we read it: “The jersey number whose player is named Bill Baulhaug equals 27.”

Or: “PlayerName inverse of Bill Baulhaug equals 27.”

 

Now, if you dare, try this with sines, cosines, and tangents:

 

 . . . Read as, “The cosine of 32 degrees equals approximately .848.”

 . . . Read as, “The angle whose cosine is .848 is approximately .”

Or: “Cosine inverse of .848 is approximately .”

 

 . . . Read as, “The tangent of 45 degrees equals 1.”

 . . . Read as “The angle whose tangent is 1 equals 45 degrees.”

Or: “Tangent inverse of 1 equals 45 degrees.”

 

 . . . Read as, “The sine of 30 degrees equals one-half.”

 . . . Read as, “The angle whose sine is one-half equals 30 degrees.”

Or: “Sine inverse of .5 equals 30 degrees.”

 

Warning: An extremely common student error is to write an expression like  or . Do you realize why both of these are nonsense? The first () is nonsense because there is no angle whose sine is 30 degrees; the sine is a ratio of opposite over hypotenuse, not an angle. The second expression () is nonsense for a different reason, namely that it is not possible for the ratio of opposite over hypotenuse to exceed 1. For example, you can correctly write  (look it up on p. 424 to make sure I’m right!), but you cannot request ; it simply can’t happen.

 

Asking for the sine inverse of 30 degrees would be immediately obvious as foolishness if you understood the Frannie the sportswriter example. The analogous question would be, “Frannie, what is PlayerName−1 (27)?” Frannie would say, “Hunh? You want to know the jersey number whose player is named 27? You’re crazy; 27 is not somebody’s name. Maybe you meant to ask me PlayerName(27), not PlayerName−1(27).”

 

In the same way, we can certainly request , but not  Do you see why?

 

Exercises for you to try (then check answers below):

 

1. PlayerName(41)

2. PlayerName(11)

3. PlayerName(“Dale Dribbler”)

4. PlayerName−1(“Dale Dribbler”)

5. PlayerName−1(18)

6.

7. , where

8.

9.

10.

11. , where 0 < x < 1

12.

13.

14.

15.

 

Scoring key:

 

0-8 correct: You’re not on the right track yet. You need to reread the Mini-Lecture from the beginning. Please read more slowly this time.

 

9-12 correct: You’re on the right track. Do you understand your mistakes, and can you learn from them? You might want to reread part of the Mini-Lecture.

 

13 or 14 correct: You’re on the right track, and you own the train.

 

15 correct: You might need to be on the STA honors track!

 

Answers (don’t peek until you’ve written down all your answers above!):

 

1. PlayerName(41) = “Pete Passer”

2. PlayerName(11) = DNE

3. PlayerName(“Dale Dribbler”) = DNE since the PlayerName function requires a jersey number as input. The player’s name is the output, not the input.

4. PlayerName−1(“Dale Dribbler”) = 18

5. PlayerName−1(18) = DNE since the PlayerName−1 function requires a player name as input. The jersey number is the output, not the input.

6.

7. , where , would be a real number greater than 1

8.

9.  = DNE (nonsense, since the inverse tangent function requires a number as input, not an angle)

10.

11. , where 0 < x < 1, would be an angle less than 45 degrees

12.

13.

14.  = DNE (nonsense, since no angle can have a cosine larger than 1)

15.  cannot be evaluated if left all by itself; the inverse cosine function must have a numeric input (see, for example, #11) in order to be evaluated