Melody

Hi Will, it is good to see you on the forum again :)

What is the range of the function (3x+1)/(x+8)

I am not sure how CPhill looked at this question. This is what I'd do.

Chris may well have an easier way to look at it  

x cannot equal -8 because the denominator cannot be equal to zero.

If I do the algebraic division here then   

\(f(x)=\frac{3x+1}{x+8}=3+\frac{-23}{x+8}\\ Now\;\;\frac{-23}{x+8}\;\;\text{cannot equal 0 so } f(x)\ne 3\\ Also\\ y=\frac{-23}{x+8}+3\\ \text{is of the from }y=\frac{a}{x-h}+k \\ \text{so it must be a hyperbola with asymptotes at x=-8 and y=3.}\)

the range Chris gave is certainly correct, I just wanted to discuss the question more. :)

1) Consider the equation 2u + v + w + x + y + z = 3. How many solutions (u , v , w , x , y , z) of non-integers does this equation have?

It has an infinite number of solutions

What is the domain of the function f(x) = sqrt6x-3 

Do you mean

\( f(x) = \sqrt6*x-3 \qquad \text {Which is what you asked}\\ domain \;\;x\in R\\~\\ or\\~\\ f(x) = \sqrt{6x}-3 \\ domain\;\;x\ge 0\qquad x\in R\)

Yes they are identical :)

\(y^x=x^y\\ log(y^x)=log(x^y)\\ xlogy=ylogx\)

I just realized that the wording in your description is erroneous.

It says

x log base y = y log base x

You need another number or variable after the word 'base'.

Or you can assume they are the same base, which is what I did and leave the word base out altogether.

\(x^2+mx+n\\ =(x+3)^2\\ =x^2+6x+9 \)

So   m=6

Hi HelperId

Look back here.  These are on the original thread.

Arr, I just foung this:

In the above-linked question, I told you I had got to part C. I meant up until part B, but then I made a situational approximation. I graphed 

and x log base y = y log base x, and I estimated the interception point to be (e,e) to a measure of precision at least 1/10,000.

Your graph looks like it is probably correct.  Yes you already know that.   :)

Ok I am finally understanding why helperid is not happy.

This is NOT a hyperbola.

I overlaid this with Heureka's graph and it is not exact, it cannot be because heureka did not use exact points but it was VERY close. This graph sure looks like half a hyperbola with the line y=x crossing it. (Positive quadrant only)

SO now what is the question again????

Part C: Find the point where the 2 parts intersect.  

Part D: Explain why the 2 parts intersect each other at the point that they do.

Part E: Find a combination of 2 equations that gives the exact same graph if possible.

Part F: Explain why the nonlinear equation had the equation that it does.

Oh there they are, I do not know the answers but I'm interested too :)

This is the graph

I'm interested in some discussion on this too    

I don't remember seeing a graph like  this before.    

Helper Id you hae reposted thit same question in all these future spots