Functions

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I would do the evaluation of f(g(f(g(1/2)))) step by step, rather than trying to write a single expression.

What is g(1/2)? It's -2(1/2) = -1, because 1/2 > 0.

What is f(-1)?

And so on.
 
I would approach this one step at a time:

f(12)=?f \left(\frac{1}{2}\right) = ?gf(12)=?g\circ f \left(\frac{1}{2}\right) = ?,
fgf(12)=?f \circ g\circ f \left(\frac{1}{2}\right) = ?,
gfgf(12)=?g \circ f \circ g\circ f \left(\frac{1}{2}\right) = ?,
 
Did you mean
View attachment 32468

I would do the evaluation of f(g(f(g(1/2)))) step by step, rather than trying to write a single expression.

What is g(1/2)? It's -2(1/2) = -1, because 1/2 > 0.

What is f(-1)?

And so on.
I didn't even notice that. However, I checked, it's the same as I have written it. Probably a mistake.
Would you start evaluating from left or right side?
 
I would approach this one step at a time:

f(12)=?f \left(\frac{1}{2}\right) = ?gf(12)=?g\circ f \left(\frac{1}{2}\right) = ?,
fgf(12)=?f \circ g\circ f \left(\frac{1}{2}\right) = ?,
gfgf(12)=?g \circ f \circ g\circ f \left(\frac{1}{2}\right) = ?,
Just realized that I was echoing @Dr.Peterson, but at least my typesetting is fancier :)
 
I didn't even notice that. However, I checked, it's the same as I have written it. Probably a mistake.
Would you start evaluating from left or right side?
Evaluate it as the notation requires. Both of us (mister fancy and I;)) showed you where to start.
 
Would you start evaluating from left or right side?
I hadn't noticed until now that @blamocur showed how to answer the wrong question! What he said would be correct if you had wanted gfgf(12)g \circ f \circ g\circ f \left(\frac{1}{2}\right); but you want fgfg(12)f \circ g \circ f\circ g \left(\frac{1}{2}\right).
 
I need reminding of the definitions of the functions. [Wish the TXT complier were better}
f(x)=2 if x1,  x if ,1<x<1,  x+1 if 1xf(x)=-2\text{ if }x\le -1,\;|x|\text{ if }, -1< x<1,\;x+1\text{ if }1\le x
g(x)=  0 if x0,2x if 0<xg(x)=\;0\text{ if }x\le0, -2^x\text{ if }0<x
Are you clear on the fact that g(12)=2 ?g\left(\tfrac{1}{2}\right)=-\sqrt{2}~?
If you are clear on that then fg(12)=2f\circ g\left(\tfrac{1}{2}\right)=-2
Can you explain this for any nN+n\in\mathbb{N}^+. ffffn(2)=2  ?\underbrace {f \circ f \circ \cdots f \circ f}_n\left( { - \sqrt 2 } \right) = -2~~?

 
I hadn't noticed until now that @blamocur showed how to answer the wrong question! What he said would be correct if you had wanted gfgf(12)g \circ f \circ g\circ f \left(\frac{1}{2}\right); but you want fgfg(12)f \circ g \circ f\circ g \left(\frac{1}{2}\right).
I got 0

1651350365615.png
 
I hadn't noticed until now that @blamocur showed how to answer the wrong question! What he said would be correct if you had wanted gfgf(12)g \circ f \circ g\circ f \left(\frac{1}{2}\right); but you want fgfg(12)f \circ g \circ f\circ g \left(\frac{1}{2}\right).
I guess I did not echo you correctly :( -- thanks for the correction.
 
Looks good me.
My only concern is that you never answered the question in my opinion!
You should have somewhere that fogofog(1/2) = 0. In my opinion this is important. I would definitely take off some credit on an exam for failing to state the answer.
 
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Looks good me.
My only concern is that you never answered the question in my opinion!
You should have somewhere that fogofog(1/2) = 0. In my opinion this is important. I would definitely take off some credit on an exam for failing to state the answer.
okay i will keep that in mind!
 
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