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 $g(c)$?

Guest Dec 6, 2018

#3**0 **

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 \(g(c)\).

Guest Dec 7, 2018

#4**+1 **

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.

Melody
Dec 8, 2018

#5**+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