Finding the transformation of a function of the form $f(x)=\\frac{mx+c}{x^2-d}$ to $af(bx)$.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "If $f(x)=\\simplify{({m}x+{c})/(x^2-{d})}$ for $x\\neq \\simplify{sqrt({d})}$, what is $\\simplify{{a}f({b}x)}$?
", "advice": "To make the transformation from $f(x)=\\simplify{({m}x+{c})/(x^2-{d})}$ to $\\simplify{{a}f({b}x)}$, we need to replace the $x$-terms in $f(x)$ with $\\simplify{{b}x}$ and then multiply that function by $\\var{a}$:
\n{advice}
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\"", "description": "", "templateType": "long string", "can_override": false}, "sol2": {"name": "sol2", "group": "Ungrouped variables", "definition": "\"\\\\[ \\\\begin{split} \\\\simplify{{a}f({b}x)} &\\\\,= \\\\var{a}\\\\left(\\\\simplify[unitFactor,!expandBrackets]{({m}({b}x)+{c})/(({b}x)^2-{d})}\\\\right) \\\\\\\\ &\\\\,= \\\\var{a}\\\\left(\\\\simplify[unitFactor]{({m*b}x+{c})/({b^2}x^2-{d})}\\\\right) \\\\\\\\&\\\\,= \\\\simplify[unitFactor]{{a*simp}({m*b/simp}x+{c/simp})/({facden}({(b^2)/facden}x^2-{d/facden}))}. \\\\end{split} \\\\]
\"", "description": "", "templateType": "long string", "can_override": false}, "simp": {"name": "simp", "group": "Ungrouped variables", "definition": "if(m*b<0 and c<0,-gcd(m*b,c),gcd(m*b,c))", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "facden": {"name": "facden", "group": "Ungrouped variables", "definition": "gcd(b^2,d)", "description": "", "templateType": "anything", "can_override": false}, "sol3": {"name": "sol3", "group": "Ungrouped variables", "definition": "\"\\\\[ \\\\begin{split} \\\\simplify{{a}f({b}x)} &\\\\,= \\\\var{a}\\\\left(\\\\simplify[unitFactor,!expandBrackets]{({m}({b}x)+{c})/(({b}x)^2-{d})}\\\\right) \\\\\\\\ &\\\\,= \\\\var{a}\\\\left(\\\\simplify[unitFactor]{({m*b}x+{c})/({b^2}x^2-{d})}\\\\right) \\\\\\\\&\\\\,= \\\\simplify[unitFactor]{{a*simp}({m*b/simp}x+{c/simp})/({facden}({(b^2)/facden}x^2-{d/facden}))} \\\\\\\\ &\\\\,=\\\\simplify[unitFactor]{{a*simp/fac}({m*b/simp}x+{c/simp})/({facden/fac}({(b^2)/facden}x^2-{d/facden}))}. \\\\end{split} \\\\]
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