Functions f and g are defined by
\(f: x \mapsto 2x - 3, \; x \in \mathbb{R},\)
\(g: x \mapsto x^2 + 4x, \; x \in \mathbb{R}.\)
- Solve the equation \(ff(x) = 11.\)
- Find the range of \(g.\)
- Find the set of values of \(x\) for which \(g(x) > 12.\)
Solution
(i) To solve \(ff(x) = 11\), first find \(f(f(x))\):
\(f(x) = 2x - 3\)
\(f(f(x)) = f(2x - 3) = 2(2x - 3) - 3 = 4x - 6 - 3 = 4x - 9\)
Set \(4x - 9 = 11\):
\(4x = 20\)
\(x = 5\)
(ii) The function \(g(x) = x^2 + 4x\) is a quadratic function. The vertex form is \(g(x) = (x + 2)^2 - 4\), with a minimum at \(x = -2\).
The minimum value is \(-4\), so the range is \(g(x) \geq -4\).
(iii) To find where \(g(x) > 12\):
\(x^2 + 4x > 12\)
\(x^2 + 4x - 12 > 0\)
Factor the quadratic: \((x - 2)(x + 6) > 0\)
The critical points are \(x = 2\) and \(x = -6\).
Test intervals: \(x < -6\), \(-6 < x < 2\), \(x > 2\).
The solution is \(x < -6\) or \(x > 2\).
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