Help me with my Math homework! (another one for 9/16/04)

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GTP_event / kevinr6287 (farming account)
I really need help on my Calculus homework. How can I find the range of this equation:
f(x)=4-square root of (x^2+4x+5)

I have the domain ( Domain of x: element of all reals) and I need to find the range. I do know that the range is y is an element of (negative infinity to 3), but I need to show my work (i used a graphing calculator).

help would be great!
 
I got it.
First, look at the inside of the sqrt.
x^2+4x+5 = x^2+4x+4+1 = (x+2)^2 + 1 (I am completing the square)

So this function has the range 1 to infinity.

The square root function is monotone, so its range is sqrt(1) to infinity, or 1 to infinity.

4-x is also a monotone function, so the final range is 4-1 to 4-infinity, or 3 to negative infinity.

Ask more questions if needed.
 
Obviously, the equation is parabolic in nature, and the negative indicates that it's open downwards. Thus, the range is defined as negative infinity to the absolute maximum (where the slope equals zero… and since it's a parabola, it'll only have one true derivative of zero). So, derive it, set it equal to zero, and bingo!

[edit]: Treed! Anyway, skip's is the shorter, but more advanced way… since you're just starting calc, I'm assuming the book/your teacher would want you to do it the way I've described. Maybe not. I dunno.
 
Sage
Obviously, the equation is parabolic in nature, and the negative indicates that it's open downwards. Thus, the range is defined as negative infinity to the absolute maximum (where the slope equals zero… and since it's a parabola, it'll only have one true derivative of zero). So, derive it, set it equal to zero, and bingo!

[edit]: Treed! Anyway, skip's is the shorter, but more advanced way… since you're just starting calc, I'm assuming the book/your teacher would want you to do it the way I've described. Maybe not. I dunno.
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But it isn't parabolic, Sage, because of the quadratic inside of the square root. However, if you can show that the function has an absolute maximum, as you did, then that works too. ;)
 
Sage
Obviously, the equation is parabolic in nature, and the negative indicates that it's open downwards. Thus, the range is defined as negative infinity to the absolute maximum (where the slope equals zero… and since it's a parabola, it'll only have one true derivative of zero). So, derive it, set it equal to zero, and bingo!

[edit]: Treed! Anyway, skip's is the shorter, but more advanced way… since you're just starting calc, I'm assuming the book/your teacher would want you to do it the way I've described. Maybe not. I dunno.
Click to expand...
Well, we are still reviewing, so we haven't even started calculus! :dopey:

Skip, won't I have to solve for x to find the range? I don't get what you mean by monotone... I don't see how you have stated to factor the x²+4x+5 by just ignoring the 4-sqrt... I know that x²+4x+5 > or = 0, but isn't that finding the domain? Do I have to use the domain to find the range?

:dunce:
 
I worked from the inside out.

The algebra I did in the first step converted x^2+4x+5 to (x+2)^2 + 1 (you can verify that they are equal by expanding (x+2)^2). Now, we know that the range (y values) of (x+2)^2 is y>0, since the square of a number is always positive. So if I add 4 to that, it will have a range of y>4, or 4 to infinity.

Monotone means that a function is always increasing or always decreasing. Sqrt(x), for example, is always increasing as x increases. 4-x, on the other hand, always decreases as x increases. sin(x) is not monotone.

Only for monotone functions can you just plug the range of the inside function into the outside function to get the range of the outside function.

So, after finding the range of x²+4x+5, I looked at the next part of the function, sqrt(some stuff).

I hope this clarifies some of it. It might be easier to explain on AIM (skip0110).
 
dude....the answer is pie...


...mmmmm.....pie....
 
skip0110
But it isn't parabolic, Sage, because of the quadratic inside of the square root. However, if you can show that the function has an absolute maximum, as you did, then that works too. ;)
Click to expand...
Doesn't the square root just expand it horizontally? (If it were just x^2, then it would "expand" it to the point of turning it into a straight line, but since it has the other junk attached, it's still slightly parabolic.) Try graphing it – I did, and maybe I didn't zoom out enough, but it sure looked parabolic to me! ;)

Event Horizon – Without actually doing the math, let me lay out a simple explanation of skip's approach:

First off, the graph will absolutely never go above y=4, since the lowest possible value for the square root section is zero (4-0)… remember that a square root can never be negative. Does that part make sense?

So, with that in mind, one can't just assume that the range is negative infinity to 4, since you can't assume that the square root will ever equal zero. That's where completing the square comes in, to basically figure out the smallest possible value for the square root section, which will yield the greatest possible value for y (the greater x becomes, the smaller y becomes, so you want to find the smallest value of x).
 
Sage
Doesn't the square root just expand it horizontally? (If it were just x^2, then it would "expand" it to the point of turning it into a straight line, but since it has the other junk attached, it's still slightly parabolic.) Try graphing it – I did, and maybe I didn't zoom out enough, but it sure looked parabolic to me! ;)
Click to expand...
First, good explanation, Sage. You would make a good TA.

Second, a parabola is defined as all the points in a plane that are equidistant from a given line and a given point (or a certain kind of conic section...). But anyway, the square root function is stretching it out, making it no longer a parabola. Just because it looks like a parabola, doesn't mean that it is one. x^4 looks kind of like a parabola, but it's not, right? ;)

In fact, as you look at very big (or very small) x values, the curve does in fact approach a straight line. (This is because the x^2 term becomes dominant over the other two terms for large x).
 
Sage
Doesn't the square root just expand it horizontally? (If it were just x^2, then it would "expand" it to the point of turning it into a straight line, but since it has the other junk attached, it's still slightly parabolic.) Try graphing it – I did, and maybe I didn't zoom out enough, but it sure looked parabolic to me! ;)

Event Horizon – Without actually doing the math, let me lay out a simple explanation of skip's approach:

First off, the graph will absolutely never go above y=4, since the lowest possible value for the square root section is zero (4-0)… remember that a square root can never be negative. Does that part make sense?

So, with that in mind, one can't just assume that the range is negative infinity to 4, since you can't assume that the square root will ever equal zero. That's where completing the square comes in, to basically figure out the smallest possible value for the square root section, which will yield the greatest possible value for y (the greater x becomes, the smaller y becomes, so you want to find the smallest value of x).
Click to expand...
That somwhat clewared it up. I'm still confused, but then skip told me that I can apply transformations to ranges, so I went from there... My brain is still frozen from summer. I am currently getting a 64% in AP Calculus... It's a lot harder than I thought. Hopefully, I can bring it up, I need scholarships. I'm almost guarenteed one $500/yr one, but I need the $5000/yr one! :D

I Finally finished the worksheet, though. My brain hurts now... :ouch: I hate ranges....
 
I'm taking introductory calc now in grade 12, and I am crapping my pants after reading this thread! We're still just doing factoring in my class!
 
Ontario has really bad schools. For example, everyone has to take French for 9 years, grades 1 through 9. I'd estimate that less than 5% of grade 10 students are fluent in French. And I am not overblowing that figure, it really is that pathetic.
 
I have another math question. This one seems really hard.

Find the Radius of a circle that is tangent to both the x-axis and y-axis and also tangent to the line x+y=12

I one point on the circle is (6,6) (midpoint of x+y=12).

If the center is at (x,y) then the Radius of the circle would be equal to the Square Root of ((6-y)²+(6-x)²). I can't find any X or Y values at all. I might need to do this a different way though. He told us to look at the geometry of a circle. I can't remember how to do that...*
 
Sage
Is the answer 6(root2)? (Which works out to about 8.48)
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Close, that is the diameter. I'd like to send a shout out to Skip for helping me solve it! :cheers:! It wasn't that hard, i just needed to use my brain!
 
So we can pass our Math course and get out of high school with high grades.
 
heero 12
Why in the firey black pits of hell do we need to know this crap anyway?
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Because it all leads to an understanding of Calculus, and Calculus is absolutely marvelous – tons and tons and tons of things in nature can be described with Calculus. It's almost dumbfounding how much the world and math are weaved and interlocked together. I think it was danoff who said that someday we'll be able to describe everything, including the universe and its beginnings, with math, and I truly believe that too.
 
I love math, and with beginning my sophomore year in high school I’m taking AH Algebra II. It’s not all that hard, but right now I am having trouble with my notes. I have found out that I write sloppy and can’t read my own hand writing. With that said I am trying to get organized.

I did great in Algebra I in the 8th grade. Making an A- which is a 94, and making 98 on the gateway which was very easy. Now the hard thing was Geometry! I hate Geometry and will always hate Geometry. I really screwed up in that class. At the beginning half of my freshman year in Geometry I was making a 99-A. At the end of the year, I have gone down to a mid C which is 83. Pathetic!

Now in Algebra II, I am making the same 83. I have failed one of my two tests, and my next test is on Monday. Along with my Biology test (don’t get me started in Biology. 73! Low D, almost failing!)

I understood some of the material you guys just did. Actually I just learned it a few weeks back.
 
In our maths class we have had NO notes whatsoever. It makes it difficult to study, I want a new teacher :(
 
rollazn
I love math, and with beginning my sophomore year in high school I’m taking AH Algebra II. It’s not all that hard, but right now I am having trouble with my notes. I have found out that I write sloppy and can’t read my own hand writing. With that said I am trying to get organized.

I did great in Algebra I in the 8th grade. Making an A- which is a 94, and making 98 on the gateway which was very easy. Now the hard thing was Geometry! I hate Geometry and will always hate Geometry. I really screwed up in that class. At the beginning half of my freshman year in Geometry I was making a 99-A. At the end of the year, I have gone down to a mid C which is 83. Pathetic!

Now in Algebra II, I am making the same 83. I have failed one of my two tests, and my next test is on Monday. Along with my Biology test (don’t get me started in Biology. 73! Low D, almost failing!)

I understood some of the material you guys just did. Actually I just learned it a few weeks back.
Click to expand...

I'm in Geometry right now. high 90! its not that hard. Biology is hard though. high C.
 
Well your just starting out. It will get harder from there. At the beginning of Geometry I made a 99- A. At the end of the semester I made a 83- C.
 
I made a 94 in geometry honors (took it sophomore year) . I'm taking Algebra II Honors now and I made an 81 (low C) on my first test and my homework grades range from 100 to 63, so i don't really know how i'm doing. The next test is Tuesday the 21st. I'm also taking AP Biology and loving every minute of it. I love Biology.
 
obviously the subsequential root of the f(x) is the co-effecent of the third largest prime number known subtract 9, unless x= the 15 decimal of pi, then the co-effecient equals the fact that i have no idea what i am talking about.
 

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