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Two functions f and g are defined on a suitable domain by f(x)=x^(2)-1 and g(x)=x^(2)+2 . a) Find a formula for f(g(x)) . b) Factorise f(g(x)) .

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Two functions f and g are defined on a suitable domain by f(x)=x^(2)-1 and g(x)=x^(2)+2 . a) Find a formula for f(g(x)) . b) Factorise f(g(x)) .

Two functions f and g are defined on a suitable domain by f(x)=x^(2)-1 and g(x)=x^(2)+2 . a) Find a formula for f(g(x)) . b) Factorise f(g(x)) .

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SethElite · Tutor for 8 years

Answer

<p> <br />a) \(f(g(x)) = x^4 + 4x^2 + 3\)<br />b) \(f(g(x)) = (x^2 + 1)(x^2 + 3)\)<br /></p>

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<p><br />1. To find a formula for \(f(g(x))\), we need to substitute \(g(x)\) into \(f(x)\). Given \(f(x) = x^2 - 1\) and \(g(x) = x^2 + 2\), we get:<br />\[f(g(x)) = (x^2 + 2)^2 - 1\]<br />Expanding the expression, we obtain:<br />\[f(g(x)) = x^4 + 4x^2 + 3\]<br /><br />2. To factorise \(f(g(x))\), we factorise the polynomial \(x^4 + 4x^2 + 3\). The factorisation is:<br />\[f(g(x)) = (x^2 + 1)(x^2 + 3)\]<br /></p>
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