Question

My Calculus professor gave us this algebra question for extra credit. The first person in the class to solve the problem gets 5% added to their FINAL GRADE! The problem;

g(x) = x^2 x =< 2
6-x x > 2

Find a single algebraic solution that satisfies both of these conditions.

I hardly understand what he is asking us to do, let alone how to solve it. What I thought I heard him say was that when x =< 2, g(x) = x^2, when x > 2, g(x) = 6-x.

Any help would be appreciated!

Answer 1

Try plotting these two equation

Answer 2

Ok, so there is a standard parabola and a line going through it intersecting at (2,4) and (-3,9).

Answer 3

x^2=6-x
x^2+x-6=0
(x+3)(x-2)
this tell us that solution is x=-3,2 but that's not enough

Answer 4

You might be waiting a while, hah. I was looking at the graph and that work and thinking that they were solutions. I am fairly terrible at intuitively working something out when I have no framework.

Answer 5

Hold on , we are gonna get you that 5% EC

Answer 6

Thanks so much! I'll work on this all night if I have to!

Answer 7

Is it because x^2 =/= 6-x?

For example, 1^2 =/= 6-1

Answer 8

I am having second thought, what did you cover in your calc class?

Answer 9

As far as I know this has nothing to do with calculus. In either case, today we covered some implicit differentiation, related rates, and continuity. He had said that only one student had ever solved this problem and that was with some help from the professor.

Answer 10

I am now sure this is a limit problem

Answer 11

Too bad you aren't a student of his or you could have made it two students. Hmm, maybe I will make it two students, but with help from you instead of him! :)

Answer 12

Here is what I think,
refer to graph: http://www.wolframalpha.com/input/?i=x^2%2C6-x&t=crmtb01
up to and including 2, we use x^2; from and not including 2, we use 6-x

Let's evaluate both equation at 2
2^2=4
6-2=4

So I think solution is 4

Answer 13

But that is the solution to the question what is the limit as x->2 for g(x).

Answer 14

I will come back to this, but I am pretty sure this is right

Answer 15

You're completely right, but as far as I know you did not answer the right question. He was not asking for the limit. He wants a single algebraic function, as opposed to its current form, which is piecewise.

Answer 16

The way he worded it was that you would get an answer like;

g(x) = x^2+4x-6 (arbitrary)

When x =< 2 g(x) = x^2
when x > 2 g(x) = 6-x

Answer 17

I will look at this tonight

Answer 18

Alright, great, thanks for your help and any other help you can provide!

Answer 19

is he just looking for the roots?
(x<=2) : x^2=0 when x=0
(x>2): 6-x=0 when x=6
the graph touches the x-axis when x=0 and x=6

Answer 20

no, one function that looks like this piecewise function

Answer 21

Yeah, what Imranmeah91 said. I really have no idea where to begin. Such a weird problem, but I guess that is why only one of his students have solved it.

Answer 22

well i'm kindof thinking we are going to see absolute value in this
y=|x-2|+4 is close looking
maybe this could be a starting point

Answer 23

y=|x-2|+4 works great for the left hand side of the graph but we need to work on the curve of the right hand side

Answer 24

you know who i think i would be awesome to solve this problem that guy that calls himself mathteacher
he likes to use technology to solve problems
hes awesome at it

Answer 25

Very nice, Myininaya! That is a great starting point. How did you figure that out, just intuition? What boggles my mind is how to get the other half of the equation without DRASTICALLY altering the first half.

Answer 26

yea i just seen it had sort of the same graph of |x|
but i'm kindof stumped i don't know if we can do this lol

Answer 27

he wants to make that piecewise function into one function

Answer 28

the way i read it is
\[
g(x) = \left\{
\begin{array}{lr}
x^2 & : x \leq 2\\
6-x & : x >2
\end{array}
\right.
\]
a piecewise function that cannot be written as a single algebraic expression so don't bother trying. do you have the question exactly as written?

Answer 29

it is clearly not one expression which is why it is piecewise. maybe the problem really is something else

Answer 30

i was kindof thinking that

Answer 31

maybe it is two inequalities to solve

Answer 32

he said only one student has been able to solve it in the past
so i guess whatever it is its not really easy

Answer 33

maybe it is a joke. this has nothing to do with calc in any case

Answer 34

and that student had the help of a professor

Answer 35

one is a line. the other is a parabola what do you want, a miracle?

Answer 36

yes

Answer 37

i will give it some thought while i go have a beer.

Answer 38

i think you are right though satellite
this is impossible

Answer 39

haha, Ill take a miracle. Yeah, I'm not sure what to tell you. Unfortunately he did not write the exact question down, only wrote that piecewise function and told us to come up with an a single algebraic function.

Answer 40

I am sending the professor an email to ensure that I have the right understanding of what answer he is looking for.

Answer 41

Quick update;

I sent my professor an email ensuring that I understood the question and to show him the progress that was made so far from this thread. Here is his response;

"You do seem to have the idea, but are you sure that [using abs(x) for absolute value of x] that your expression, abs(x-2) + 4, is algebraic? Can you really write it solely in terms of add, subtract, multiply, divide, power, and root? Because that’s what “algebraic” means. Now in fact that function you wrote is algebraic, but maybe as a first challenge you should actually write such a formula for it."

So there must be some way to write the function in terms of ASMDPR and this is the path I am going to pursue now, which will hopefully give me a better understanding of how to alter the function y = |x-2| + 4

Answer 42

I am gonna play with Mathematica to see if I can generate such function

Answer 43

Wow, thought I might try to do the same until I realized it has a $300 price tag!

Answer 44

ok joe we have a piecewise function
g(x)= x^2 for x>=2 and 6-x for x<2
the goal is to write using one algebra function

Answer 45

i dont see how it is possible but it seems the professor knows it is

Answer 46

I sure hope this professor is not trying to make a point

Answer 47

I don't think he would respond to my email in such a fashion if he was just trying to trick us. Plus, this is a calculus class. If he was going to trick us for extra credit I doubt it would be an algebra problem.

Answer 48

the closer thing i could come up with is y=|x-2|+4

Answer 49

yeah, i drew a rough picture of it but nothing is coming to mind immediately. Do we know what the student has learned right before this was asked? like what the topic is over or something?

Answer 50

Yeah, we were covering related rates, continuity, and implicit differentiation. He used this to prove some point about continuity (I believe), but the extra credit question had nothing to do with what we were learning.

He did say that only one student had ever solved this problem and that was with some help from the professor himself.

Answer 51

or maybe it would be an algebra problem since we are supposedly to know algebra

Answer 52

What is the equation for that line, imranmeah91?

Answer 53

i believe no pattern exist for the number sequence
6,5,4,9,16,... does it?

hey joe can you come up with an expression like you did that one time for that one pattern that no one else could get

Answer 54

How about this?

Answer 55

i can create a formula for that sequence, but i dont know if that would help, i dont have that much control over the tail of the sequence.

Answer 56

i wish there what a natural sequence that what like, u(t) = 0 for x <0, and 1 for x>= 0, that would solve this in a heart beat.

Answer 57

Did you see my graph?

Answer 58

:( ok don't waste your time then

Answer 59

how did you get that imranmeah?

Answer 60

Yeah imran, what is the formula for that graph? Looks damn close.

Answer 61

He used mathematica I believe

Answer 62

I used mathematica, it is not exact but you may be able to adjust

Answer 63

it needs to be shifted to the right some.

Answer 64

New graph and equation hold on

Answer 65

im posting a (crude) pic of what the function looks like.

Answer 66

just follow the solid lines.

Answer 67

\[4-\sqrt[3]{(2-x)^2} (x+6)^{2/3}\]

Answer 68

I think it need to be adjusted from here

Answer 69

the best thing i can think of here is to use an indicator funtion

Answer 70

oops i have been doing it for x^2,x>=2 and 6-x, x<2

Answer 71

let g(x)=x^2 * I where I=1 when x<=2 and I=(6-x)/x^2 when x>2...but it seems simple

Answer 72

Imranmeah, after putting that function into my graphing calculator I one problem in that anything anything < -6 is not a part of the parabola.

When x = -8 y should be 64, but with that function it is -3.36.

Damn good start though.

Answer 73

i think i might be on to something. Check this out:
http://www.wolframalpha.com/input/?i=%281%2F%284-2x%29%29%5B |2-x|%2B%282-x%29%5Dx^2%2B%281%2F%282x-4%29%29%5B|x-2|%2B%28x-2%29%5D%286-x%29

Answer 74

That's perfect

Answer 75

this was my idea:
I want to make some function like:
\[h(x) = ax^2+b(6-x)\]
where a = 1, b = 0 when x is less than 2, and a = 0, b = 1 when x is greater than 2.

Answer 76

so the trouble comes in finding some functions like that, but what i ended up using at first is:
a =|2 - x| + (2 - x)
b = |x - 2| + (x - 2)
because when x is less than 2, b will equal 0, and when x is greater than 2, a will equal 0.

Answer 77

thats the same idea as trying to make an indicator function, where ur indicators a nd b take on different values at different values of x

Answer 78

My god, Joe, brilliant. I have yet to plug the numbers in to check, but this looks great so far.

Answer 79

but their values arent 1 when their respective conditions are satisfied:
if x is less than 2, a becomes:
2(2 - x) = 4 - 2x
and if x is greater than 2, b becomes:
2(x - 2) = 2x-4
So i just divide by those values to make the constant come out to 1 thus:
\[a = \frac{1}{4-2x}(|2-x|+(2-x)), b = \frac{1}{2x-4}(|x-2|+(x-2))\]
and the equation you are looking for is:
\[g(x) = ax^2+b(6-x)\]

Answer 80

Joe, when plugging the number 2 into the formula, which should give an answer of four, came out indeterminate. Any idea why?

Answer 81

because of my fractions >.< let me see if i can think of some way to take care of that.

Answer 82

1/(4-2x) = 1/0 when x=2

Answer 83

i got nothing, i dont see anyway to avoid the discontinuity at x = 2 =/ other than that, does it seem good? anyone have any objections to that?

Answer 84

I think it is a good as it is gonna get

Answer 85

wow joe that is amazing!

Answer 86

that is my cup of joe
i dont share

Answer 87

but i so have one small (very very small) criticism namely that you are using absolute value

Answer 88

Yeah, that really was amazing Joe. Absolutely brilliant. I think I might have seen you on Good Will Hunting ;)

Answer 89

which itself is piecewise.

Answer 90

shh satellite

Answer 91

^ Although my professor did note that it was acceptable to use it.

Answer 92

sorry i go back to the dungeon now

Answer 93

thanks :) i got the idea when myininaya posted that abs function. so thank you myininaya! and now ive learned something new, yay :) thats why i like this site lol.

Answer 94

gj joe :)

Answer 95

i learn something here every day (almost)