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The graph of $y=f(x)$ is shown below. Assume $f(x)$ is defined only on the domain shown, and that the $y$-axis is an axis of symmetry for the graph.

For any real number $c$, let $g(c)$ be the number of solutions to the equation $f(x)=c$. What is the average of all distinct values in the range of $g(c)$?

 

 

 Dec 6, 2018
 #1
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anyone?

 Dec 7, 2018
 #2
avatar+109509 
0

Could you please edit your question to get rid of all the irrelevant $ signs.

Then it will be easier to read and someone might attempt an answer.

Melody  Dec 7, 2018
 #3
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The graph of \(y=f(x)\) is shown below. Assume \(f(x)\) is defined only on the domain shown, and that the \(y\)-axis is an axis of symmetry for the graph.

For any real number \(c\), let \(g(c)\) be the number of solutions to the equation \(f(x) = c\). What is the average of all distinct values in the range of \(g(c)\)

 

 Dec 7, 2018
 #4
avatar+109509 
+2

I think questions like this one are always really hard to understand and I would like another mathematician to check my answer.

 

 

Use my pic, which is near enough to the same.

 

Think about a horizontal line through the graph.  

At the top it will pass through the graph twice so  g(c)=2

 

Here is a pic

 

I have drawn six horizonal lines.

 

For the top (purple - 1st ) one    f(x)=-2

this crosses the graph f(x) at 2 different x values so  g(-2)=2

 

the next horizontal line (black 2nd )  palsses through 4 points so  g(-2.8)=4

 

the 3rd horizontal line (red )  palsses through 6 points so  g(-3.4)=6

 

the 4th horizontal line (blue )  palsses through 5 points so  g(-4)=5

 

the 5th horizontal line (green)  palsses through 4 points so  g(-5)=4

 

the bottom horizontal line (purple )  palsses through 2 points so  g(-6.7)=2

 

So g(c) can be 2, 4, 5, or 6       

 

The average of the distict values of g(c) = (2+4+5+6)/4 = 17/4 = 4.25

 

I am pretty sure that is correct.

 

 

Here is the graph I used         https://www.desmos.com/calculator/ft9bo78w4l       

You do not need it though.

 

 

 

 

 Dec 8, 2018
edited by Melody  Dec 8, 2018
 #5
avatar+2419 
+1

Since you wanted another mathematician to check this, I will say that this is how I interpreted the question as well. I believe you got the correct answer, Melody. 

TheXSquaredFactor  Dec 8, 2018
 #6
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Thanks x-squared :)

Melody  Dec 9, 2018
 #7
avatar+28 
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blush, sorry but the right answer is 3.4

otaku  Dec 12, 2018
 #8
avatar+109509 
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Yes I forgot the 0.

No big deal, I showed you how to do it. 

You can pick up trivial errors like that on your own.

 

You do not need to be sorry - I did it correctly. 

A thank you would be nice though.

Melody  Dec 12, 2018

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