Rojas Lindsey
01/28/2024 · Middle School
Functions \( f, \mathrm{~g} \) and h are defined as follows: \[ \mathrm{f}: x \mapsto x-4 x^{\frac{1}{2}}+1 \quad \text { for } x \geqslant 0, \] \( \mathrm{~g}: x \mapsto m x^{2}+n \) for \( x \geqslant-2 \), where \( m \) and \( n \) are constants, \( \mathrm{h}: x \mapsto x^{\frac{1}{2}}-2 \) for \( x \geqslant 0 \). (a) Solve the equation \( \mathrm{f}(x)=0 \), giving your solutions in the form \( x=a+b \sqrt{c} \), where \( a, b \) and \( c \) are integers. (b) Given that \( \mathrm{f}(x)=\mathrm{gh}(x) \), find the values of \( m \) and \( n \).
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(a) The solutions to \( \mathrm{f}(x) = 0 \) are \( x = 7 + 4\sqrt{3} \) and \( x = 7 - 4\sqrt{3} \).
(b) There is no solution for \( m \) and \( n \) that satisfies \( \mathrm{f}(x) = \mathrm{gh}(x) \) for all \( x \geqslant 0 \).
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