MHB Is the Composition of Functions $F$ and $G$ Correct?

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The discussion centers on the composition of functions F and G, where F(x) = x + 5 and G(x) = |x|/x for x ≠ 0, with G(0) = 1. There is a debate about the correctness of the composition G(F(x)), with one participant asserting that the definition leads to a conflict when evaluating G at certain points, suggesting it may not define a proper function. Another participant agrees with the initial composition but recommends sticking to the original definitions rather than breaking them down into cases. The conversation concludes with a request for clarification on the graphing commands used.
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$F(x) = x + 5\qquad\qquad G(x) = \frac{|x|}{x}, \ \text{if} \ x\neq 0, \ G(0) = 1$

$G(F(x)) = G(x + 5) = \frac{|x + 5|}{x + 5} = \begin{cases}
1 & \text{if} \ x \geq 0\\
-1 & \text{if} \ x\in (0,-5)\cup (-5,\infty)
\end{cases}$

Is the correct for the composition?
 
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I'm not sure I'd quite agree yet. I would do
$$G(F(x))=G(x+5)=\begin{cases}\frac{|x+5|}{x+5}, \quad & x+5\not=0\\
1, \quad & x+5=0\end{cases},$$
and go from there.
 
Ackbach said:
I'm not sure I'd quite agree yet. I would do
$$G(F(x))=G(x+5)=\begin{cases}\frac{|x+5|}{x+5}, \quad & x+5\not=0\\
1, \quad & x+5=0\end{cases},$$
and go from there.

View attachment 325
I just graphed it and it looks right.
 
I agree with Ackbach. Perhaps it would be better if you just worked with the definition of $G$ as given instead of breaking it into exact expressions.
 
dwsmith said:
$F(x) = x + 5\qquad\qquad G(x) = \frac{|x|}{x}, \ \text{if} \ x\neq 0, \ G(0) = 1$

$G(F(x)) = G(x + 5) = \frac{|x + 5|}{x + 5} = \begin{cases}
1 & \text{if} \ x \geq 0\\
-1 & \text{if} \ x\in (0,-5)\cup (-5,\infty)
\end{cases}$

Is the correct for the composition?

Your definition has a conflict in it. Suppose $x=1$. Then it satisfies $x\geq 0$ as well as being in the interval $(-5,\infty)$. So $G \circ F$ would evaluate both to $+1$ and $-1$. Therefore, the composition you have defined there is not a function, but a relation. Either that, or it's an ill-defined function.
 
dwsmith said:
View attachment 325
I just graphed it and it looks right.

What exact commands did you execute to produce this graph?
 
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