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# Find $gof$ and $fog$, if $f(x) = 8x^3$ and $g(x)=x^\frac{1}{3}$

Note: This is part 2 of a 2 part question, split as 2 separate questions here.

Toolbox:
• Given two functions $f:A \to B$ and $g:B \to C$, then composition of $f$ and $g$, $gof:A \to C$ by $gof (x)=g(f(x))\;for\; all \;x \in A$
• Given two functions $g:A \to B$ and $f:B \to C$, then composition of $g$ and $g$, $fog:A \to C$ by $fog (x)=f(g(x))\;for\; all \;x \in A$
Given $f(x)=8x^3 \;and\;g(x)=x^{1/3}$
Step 1: Given two functions $f:A \to B$ and $g:B \to C$, then composition of $f$ and $g$, $gof:A \to C$ by $gof (x)=g(f(x))\;for\; all \;x \in A$
Since $f(x) = 8x^3 \rightarrow (gof) (x)=g(f(x)) = g(8 x^3) = (8x^3)^{1/3} = 2x$
Step 2: Given two functions $g:A \to B$ and $f:B \to C$, then composition of $g$ and $g$, $fog:A \to C$ by $fog (x)=f(g(x))\;for\; all \;x \in A$
Since $g(x)=x^{1/3} \rightarrow fog(x) = f(g(x)) = f (x^{1/3}) = 8(x^{1/3}) = 8x$
edited Mar 19, 2013