Evaluating composite functions involving a linear function and a modulus function of the form $f(x)=|x|+c$, for a given value of $x$.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "If $f(x)=\\simplify{abs(x)+{c}}$ and $g(x)=\\simplify{{n}x+{d}}$, calculate $f\\circ g(x)$ and $g \\circ f(x)$ when $x=\\var{a}$.
\n\nRecall: $f \\circ g(x) \\equiv f(g(x))$ and $g \\circ f(x) \\equiv g(f(x))$.
", "advice": "\nTo find the composition $f \\circ g(x)$ we are substituting the expression for $g(x)$ into the function $f(x)$, replacing the $x$-terms with the function $g(x)$. Similarly, to find the composition $g \\circ f(x)$ we are substituting the expression for $f(x)$ into the function $g(x)$, replacing the $x$-terms with the function $f(x)$.
\nSo, for $f(x)=\\simplify{abs(x)+{c}}$ and $g(x)=\\simplify{{n}x+{d}}$,
\n\\[ f \\circ g(x) \\equiv f(g(x)) = \\simplify{abs({n}x+{d})+{c}},\\]
\nand
\n\\[ \\begin{split} g \\circ f(x) \\equiv g(f(x)) &\\,= \\simplify{{n}(abs(x)+{c})+{d}} \\\\ &\\,=\\simplify[!collectNumbers,unitFactor]{{n}abs(x)+{n*c}+{d}} \\\\ &\\,=\\simplify{{n}abs(x)+{n*c+d}}. \\end{split} \\]
\nWhen $x=\\var{a}$,
\n\\[ \\begin{split} f \\circ g(\\var{a}) \\equiv f(g(\\var{a})) &\\,= \\simplify[alwaysTimes]{abs({n}{a}+{d})+{c}} \\\\ &\\,=\\simplify{abs({n*a}+{d})+{c}} \\\\ &\\,=\\simplify[!collectNumbers]{{abs(n*a+d)}+{c}}\\\\ &\\,=\\simplify{{abs(n*a+d)}+{c}},\\end{split}\\]
\nand
\n\\[ \\begin{split} g \\circ f(\\var{a}) \\equiv g(f(\\var{a})) &\\,=\\simplify{{n}abs({a})+{n*c+d}} \\\\ &\\,=\\simplify[!collectNumbers]{{n}{abs(a)}+{n*c+d}} \\\\ &\\,=\\simplify[!collectNumbers]{{n*abs(a)+(n*c+d)}} . \\end{split} \\]
\n\nAlternative Method:
\nRather than finding the composite functions first and then evaluating them, we can calculate $g(\\var{a})$ and $f(\\var{a})$, and then substitute those values into the other functions:
\n\\[ \\begin{split} g(\\var{a}) &\\,=\\simplify[alwaysTimes]{{n}{a}+{d}} \\\\ &\\,=\\simplify{{n*a}+{d}}, \\end{split} \\]
\nhence
\n\\[ \\begin{split} f \\circ g(\\var{a}) = f(\\simplify{{n*a+d}}) &\\,= \\simplify{abs({n*a+d})+{c}} \\\\ &\\,= \\simplify{{abs(n*a+d)+c}}. \\end{split} \\]
\nSimilarly,
\n\\[ \\begin{split} f(\\var{a}) &\\,= \\simplify{abs({a})+{c}}\\\\ &\\,=\\simplify[!collectNumbers]{{abs(a)}+{c}} \\\\ &\\,=\\simplify{{abs(a)+c}} \\end{split} \\]
\nhence,
\n\\[\\begin{split} g \\circ f(\\var{a}) = g(\\simplify{{abs(a)+c}}) &\\,= \\simplify[alwaysTimes]{{n}{abs(a)+c}+{d}} \\\\ &\\,=\\simplify{{n}{abs(a)+c}+{d}}.\\end{split} \\]
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\n$g \\circ f(\\var{a})=$[[1]]
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