Need help with some equations (rational Functions)

NichKrol

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Oct 21, 2019
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6
1. Solve the equation & state restrictions:
(1/x-4)+(x/x-2)=(2/x^2-6x+8)

2.Create a simplified function that represents the sum of an unknown
positive number x and its reciprocal.

3. Given that the point (1, -1/3) is on this graph, create the equation for the rational function
shown using what you know about rational functions.
Annotation 2019-10-21 134416.png
(The graph is attached, If you are capable, help me solve whatever you can. Godspeed.
 
What do you need HELP with? Where are you stuck? What have you tried? We need something from you to work with.

in 2, if you chose x to be a number then what is the reciprocal of that number?
 
What do you need HELP with? Where are you stuck? What have you tried? We need something from you to work with.

in 2, if you chose x to be a number then what is the reciprocal of that number?
the regular form of x if you treated it as a number would be x/1, so the reciprocal would be 1/x. I've tried a bunch of things but I dont think any of it is right and I need to see where other people start out for these problems at least.
 
the regular form of x if you treated it as a number would be x/1, so the reciprocal would be 1/x. I've tried a bunch of things but I dont think any of it is right and I need to see where other people start out for these problems at least.
Please share some of those bunch of things - so that we know where exactly you are going wrong (if you are going wrong) and how we can correct your way.
 
Please share some of those bunch of things - so that we know where exactly you are going wrong (if you are going wrong) and how we can correct your way.
My problem is that ive been attempting them for a while and my brain has hit a complete standstill- The reason I'm here is to get another persons perspective on how theyre going about this instead of someone telling me where I went wrong. I already know I'm incorrect. one I tried is
Y=X+1/X, with the reciprocal being Y=X+X/1
 
My problem is that ive been attempting them for a while and my brain has hit a complete standstill- The reason I'm here is to get another persons perspective on how theyre going about this instead of someone telling me where I went wrong. I already know I'm incorrect. one I tried is
Y=X+1/X, with the reciprocal being Y=X+X/1
Actually, the benefit of a site like this is exactly that we can help you correct what you are doing wrong, whereas all a textbook can do is to show you how someone else solves a similar problem. If you want the latter, look in your book! We want to actually help, not just show solutions as a textbook would do.

As for this problem, #2, you have given the correct answer, y = x + 1/x. This is the sum of a number x and its reciprocal, 1/x. Possibly the answer should be written as f(x) rather than y.

But when you say, "the reciprocal being y=x+x/1", it looks like you might have a misconception or two. You are not being asked to find the reciprocal of the expression you wrote; and if you were, it would be wrong to take the reciprocal of each term separately. So, what are you trying to say there?

Honestly, you have to believe us when we say that showing your work will allow us to give you better help. We do more than just tell you you're wrong (as the back of a book can do).
 
found common denominators, combined numerators, factored with AC and simplified, the only true solution i found is x=-1. Still feels wrong. Thoughts?
You can check that

First, by putting x = -1 to the left-hand-side of the equation and calculate a "value". Then:

by putting x = -1 to the right-hand-side of the equation and calculate a "value". If these two "value"s turn out to be equal - you are most probably "correct". If those turn out to be different - then you made a mistake in one of those steps.

Please share your work "numerically" as opposed to "in words".
 
found common denominators, combined numerators, factored with AC and simplified, the only true solution i found is x=-1. Still feels wrong. Thoughts?
Why does it "feel wrong"? What specifically are you unsure of in your work, or doubtful of in the answer?
 
My problem is that ive been attempting them for a while and my brain has hit a complete standstill- The reason I'm here is to get another persons perspective on how theyre going about this instead of someone telling me where I went wrong. I already know I'm incorrect. one I tried is
Y=X+1/X, with the reciprocal being Y=X+X/1
OK the sum of a positive number and its reciprocal can indeed be expressed as the function

[MATH]f(x) = x + \dfrac{1}{x}.[/MATH]
Because you are studying rational functions, they may want that expressed in a different form as

[MATH]f(x) = x + \dfrac{1}{x} = \dfrac{x}{1} + \dfrac{1}{x} = \dfrac{x^2}{x} + \dfrac{1}{x} = \dfrac{x^2 + 1}{x}.[/MATH]
 
1. Solve the equation & state restrictions:
(1/x-4)+(x/x-2)=(2/x^2-6x+8)


You have grouping symbols in the wrong place.

1/(x - 4) + x/(x - 2) = 2/(x^2 - 6x + 8)

or

\(\displaystyle \dfrac{1}{x - 4} \ + \ \dfrac{x}{x - 2} \ = \ \dfrac{2}{x^2 - 6x + 8}\)
 
found common denominators, combined numerators, factored with AC and simplified, the only true solution i found is x=-1. Still feels wrong. Thoughts?
Your "complete" answer should also include restriction on "x".
 
To work out restrictions on x for the equation:
1/(x - 4) + x/(x - 2) = 2/[(x - 2)(x - 4)]
determine values of x which make any of the fractions undefined (zero denominator):
x - 4 = 0 when x =
₋₋₋₋₋₋₋₋₋₋
x - 2 = 0 when x = ₋₋₋₋₋₋₋₋₋₋
State restrictions on x:
x ≠
₋₋₋₋₋₋₋₋₋₋ or x ≠ ₋₋₋₋₋₋₋₋₋₋
Write restrictions on x at each step of your solution. Your final solution to the quadratic equation x² - 3x - 4 = 0 yields two answers - only one of these (the same answer you came up with) is possible. State clearly why the other answer is not possible in this case. You can demonstrate this by drawing graphs of f(x) = 1/(x - 4) + x/(x - 2) and g(x) = 2/[(x - 2)(x - 4)] on the same set of axis. Find at what x value the curves intersect and why they cannot intersect at the second x value found in your solution to x² - 3x - 4 = 0.
 
Graph 1: y = 1/(x - 4) + x/(x - 2)
Divide x - 2 into x since power of x in numerator is ≥ power of x in denominator:

C4B2968C-0D0F-4695-A875-C7220A6669DA.jpeg

So the rule of graph 1 becomes: y = 1/(x - 4) + 2/(x - 2) + 1
Curve has vertical asymptotes at x = 2 and x = 4 (y approaches infinity as x approaches 2 from above and 4 from below) and a horizontal asymptote at y = 1 (y approaches 1 as x approaches positive or negative infinity).

Graph 2: y = 2/[(x - 2)(x - 4)]
Curve has vertical asymptotes at x = 2 and x = 4 (as x approaches 4 y approaches positive or negative infinity, as x approaches 2 y approaches positive or negative infinity) and a horizontal asymptote at y = 0 (y approaches the X axis as x approaches positive or negative infinity).

Curves intersect when x = - 1 ( the solution to the equation in question 1 - formed by equating the two graphs) :
1/(x - 4) + x/(x - 2) = 2/[(x - 2)(x - 4)])
Find y value of point of intersection by substituting x = -1 into y = 2/[(x - 2)(x - 4)]:
y = 2/[(- 1 - 2)(- 1 - 4)]
= 2/( - 3 x - 5)
= 2/15
Therefore curves intersect at the point (-1, 2/15).
 

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Hints for question 3:

Which curve in question 1 looks like the curve in question 3 but slightly shifted?
What does the denominator tell us about any vertical asymptotes for the curve?
Consider graph of the form y = a/[(x - b)(x - c)] + d which passes through (2, 1)
and has asymptotes at x = -1, x = 5, y = 2.
It follows: b = -1, c = 5, d = 2
We have y = a/[(x + 1)(x - 5)] + 2
Substituting x = 2, y = - 7 gives:
1 = a/[(2 + 1)(2 - 5)] + 2
1 = a/(3 x -3) + 2
-1 = a/-9
a = 9
So curve has rule y = 9/[(x + 1)(x - 5)] + 2

F5D8B399-D3D8-43C6-9203-61724A432A41.jpeg
Hope this example helps.
 
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