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For example if $f(x) = x + 2$ and $g(x) = 4x - 1$

Then what would be the difference in $f \cdot g$ and $f(g(x))$?

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If the dot is meant to be the composite function then they are the same thing – John Jan 2 '14 at 16:40

The notation $f \cdot g$ means that for every $x$ the function is $$ (f \cdot g)(x) = f(x) \cdot g(x) $$ which is pointwise multiplication.

On the other hand $f \circ g$ is the composition of functions, $$ (f \circ g)(x) = f(g(x)) \ . $$

For your examples: $$ f(x) \cdot g(x) = (x+2) \cdot (4x-1) = 4x^2 + 8x - x - 2 = 4x^2 + 7x -2 $$ while $$ (f \circ g)(x) = f(g(x)) = f(4x-1) = (4x-1)+2 = 4x + 1 \ . $$

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If dot means composition then they are the same. $$(f \cdot g)(x) = f(g(x))=(4x-1)+2=4x+1$$ If dot means multiplication then $$(f \cdot g)(x)=(x+2)(4x-1)=4x^2+7x-2$$

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If your interpretation of $\cdot$ is $\circ$ as in $f \circ g(x)$ Then, $f \circ g(x)=f(g(x))$ which is a composite function. Note that $(f \circ g)(x) \neq (g \circ f)(x)$

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